TY - JOUR
AB - We continue our study of ‘no‐dimension’ analogues of basic theorems in combinatorial and convex geometry in Banach spaces. We generalize some results of the paper (Adiprasito, Bárány and Mustafa, ‘Theorems of Carathéodory, Helly, and Tverberg without dimension’, Proceedings of the Thirtieth Annual ACM‐SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, San Diego, California, 2019) 2350–2360) and prove no‐dimension versions of the colored Tverberg theorem, the selection lemma and the weak 𝜀 ‐net theorem in Banach spaces of type 𝑝>1 . To prove these results, we use the original ideas of Adiprasito, Bárány and Mustafa for the Euclidean case, our no‐dimension version of the Radon theorem and slightly modified version of the celebrated Maurey lemma.
AU - Ivanov, Grigory
ID - 9037
JF - Bulletin of the London Mathematical Society
SN - 00246093
TI - No-dimension Tverberg's theorem and its corollaries in Banach spaces of type p
ER -
TY - JOUR
AB - We study properties of the volume of projections of the n-dimensional
cross-polytope $\crosp^n = \{ x \in \R^n \mid |x_1| + \dots + |x_n| \leqslant 1\}.$ We prove that the projection of $\crosp^n$ onto a k-dimensional coordinate subspace has the maximum possible volume for k=2 and for k=3.
We obtain the exact lower bound on the volume of such a projection onto a two-dimensional plane. Also, we show that there exist local maxima which are not global ones for the volume of a projection of $\crosp^n$ onto a k-dimensional subspace for any n>k⩾2.
AU - Ivanov, Grigory
ID - 9098
IS - 5
JF - Discrete Mathematics
SN - 0012365X
TI - On the volume of projections of the cross-polytope
VL - 344
ER -
TY - JOUR
AB - Let A={A1,…,An} be a family of sets in the plane. For 0≤i2b be integers. We prove that if each k-wise or (k+1)-wise intersection of sets from A has at most b path-connected components, which all are open, then fk+1=0 implies fk≤cfk−1 for some positive constant c depending only on b and k. These results also extend to two-dimensional compact surfaces.
AU - Kalai, Gil
AU - Patakova, Zuzana
ID - 7960
JF - Discrete and Computational Geometry
SN - 01795376
TI - Intersection patterns of planar sets
VL - 64
ER -
TY - CONF
AB - We prove general topological Radon-type theorems for sets in ℝ^d, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly theorem, and consequently the existence of weak ε-nets as well as a (p,q)-theorem. More precisely: Let X be either ℝ^d, smooth real d-manifold, or a finite d-dimensional simplicial complex. Then if F is a finite, intersection-closed family of sets in X such that the ith reduced Betti number (with ℤ₂ coefficients) of any set in F is at most b for every non-negative integer i less or equal to k, then the Radon number of F is bounded in terms of b and X. Here k is the smallest integer larger or equal to d/2 - 1 if X = ℝ^d; k=d-1 if X is a smooth real d-manifold and not a surface, k=0 if X is a surface and k=d if X is a d-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let F be a finite family of open sets in a surface S such that the intersection of any subfamily of F is either empty, or path-connected. Then the fractional Helly number of F is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a (p,q)-theorem for open subsets of a surface.
AU - Patakova, Zuzana
ID - 7989
SN - 18688969
T2 - 36th International Symposium on Computational Geometry
TI - Bounding radon number via Betti numbers
VL - 164
ER -
TY - CONF
AB - Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation on P is a full triangulation of some subset P' of P containing all extreme points in P. A bistellar flip on a partial triangulation either flips an edge, removes a non-extreme point of degree 3, or adds a point in P ⧵ P' as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The goal of this paper is to investigate the structure of this graph, with emphasis on its connectivity. For sets P of n points in general position, we show that the bistellar flip graph is (n-3)-connected, thereby answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos) on triangulations. This matches the situation for the subfamily of regular triangulations (i.e., partial triangulations obtained by lifting the points and projecting the lower convex hull), where (n-3)-connectivity has been known since the late 1980s through the secondary polytope (Gelfand, Kapranov, Zelevinsky) and Balinski’s Theorem. Our methods also yield the following results (see the full version [Wagner and Welzl, 2020]): (i) The bistellar flip graph can be covered by graphs of polytopes of dimension n-3 (products of secondary polytopes). (ii) A partial triangulation is regular, if it has distance n-3 in the Hasse diagram of the partial order of partial subdivisions from the trivial subdivision. (iii) All partial triangulations are regular iff the trivial subdivision has height n-3 in the partial order of partial subdivisions. (iv) There are arbitrarily large sets P with non-regular partial triangulations, while every proper subset has only regular triangulations, i.e., there are no small certificates for the existence of non-regular partial triangulations (answering a question by F. Santos in the unexpected direction).
AU - Wagner, Uli
AU - Welzl, Emo
ID - 7990
SN - 18688969
T2 - 36th International Symposium on Computational Geometry
TI - Connectivity of triangulation flip graphs in the plane (Part II: Bistellar flips)
VL - 164
ER -
TY - CONF
AB - We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call homotopic curve shortening (HCS), starts with a closed curve (which might self-intersect) in the presence of a set P⊂ ℝ² of point obstacles, and evolves in discrete steps, where each step consists of (1) taking shortcuts around the obstacles, and (2) reducing the curve to its shortest homotopic equivalent. We find experimentally that, if the initial curve is held fixed and P is chosen to be either a very fine regular grid or a uniformly random point set, then HCS behaves at the limit like the affine curve-shortening flow (ACSF). This connection between HCS and ACSF generalizes the link between "grid peeling" and the ACSF observed by Eppstein et al. (2017), which applied only to convex curves, and which was studied only for regular grids. We prove that HCS satisfies some properties analogous to those of ACSF: HCS is invariant under affine transformations, preserves convexity, and does not increase the total absolute curvature. Furthermore, the number of self-intersections of a curve, or intersections between two curves (appropriately defined), does not increase. Finally, if the initial curve is simple, then the number of inflection points (appropriately defined) does not increase.
AU - Avvakumov, Sergey
AU - Nivasch, Gabriel
ID - 7991
SN - 18688969
T2 - 36th International Symposium on Computational Geometry
TI - Homotopic curve shortening and the affine curve-shortening flow
VL - 164
ER -
TY - CONF
AB - Let K be a convex body in ℝⁿ (i.e., a compact convex set with nonempty interior). Given a point p in the interior of K, a hyperplane h passing through p is called barycentric if p is the barycenter of K ∩ h. In 1961, Grünbaum raised the question whether, for every K, there exists an interior point p through which there are at least n+1 distinct barycentric hyperplanes. Two years later, this was seemingly resolved affirmatively by showing that this is the case if p=p₀ is the point of maximal depth in K. However, while working on a related question, we noticed that one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample; this re-opens Grünbaum’s question. It follows from known results that for n ≥ 2, there are always at least three distinct barycentric cuts through the point p₀ ∈ K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through p₀ are guaranteed if n ≥ 3.
AU - Patakova, Zuzana
AU - Tancer, Martin
AU - Wagner, Uli
ID - 7992
SN - 18688969
T2 - 36th International Symposium on Computational Geometry
TI - Barycentric cuts through a convex body
VL - 164
ER -
TY - CONF
AB - In the recent study of crossing numbers, drawings of graphs that can be extended to an arrangement of pseudolines (pseudolinear drawings) have played an important role as they are a natural combinatorial extension of rectilinear (or straight-line) drawings. A characterization of the pseudolinear drawings of K_n was found recently. We extend this characterization to all graphs, by describing the set of minimal forbidden subdrawings for pseudolinear drawings. Our characterization also leads to a polynomial-time algorithm to recognize pseudolinear drawings and construct the pseudolines when it is possible.
AU - Arroyo Guevara, Alan M
AU - Bensmail, Julien
AU - Bruce Richter, R.
ID - 7994
SN - 18688969
T2 - 36th International Symposium on Computational Geometry
TI - Extending drawings of graphs to arrangements of pseudolines
VL - 164
ER -
TY - THES
AB - Algorithms in computational 3-manifold topology typically take a triangulation as an input and return topological information about the underlying 3-manifold. However, extracting the desired information from a triangulation (e.g., evaluating an invariant) is often computationally very expensive. In recent years this complexity barrier has been successfully tackled in some cases by importing ideas from the theory of parameterized algorithms into the realm of 3-manifolds. Various computationally hard problems were shown to be efficiently solvable for input triangulations that are sufficiently “tree-like.”
In this thesis we focus on the key combinatorial parameter in the above context: we consider the treewidth of a compact, orientable 3-manifold, i.e., the smallest treewidth of the dual graph of any triangulation thereof. By building on the work of Scharlemann–Thompson and Scharlemann–Schultens–Saito on generalized Heegaard splittings, and on the work of Jaco–Rubinstein on layered triangulations, we establish quantitative relations between the treewidth and classical topological invariants of a 3-manifold. In particular, among other results, we show that the treewidth of a closed, orientable, irreducible, non-Haken 3-manifold is always within a constant factor of its Heegaard genus.
AU - Huszár, Kristóf
ID - 8032
SN - 2663-337X
TI - Combinatorial width parameters for 3-dimensional manifolds
ER -
TY - THES
AB - We present solutions to several problems originating from geometry and discrete mathematics: existence of equipartitions, maps without Tverberg multiple points, and inscribing quadrilaterals. Equivariant obstruction theory is the natural topological approach to these type of questions. However, for the specific problems we consider it had yielded only partial or no results. We get our results by complementing equivariant obstruction theory with other techniques from topology and geometry.
AU - Avvakumov, Sergey
ID - 8156
TI - Topological methods in geometry and discrete mathematics
ER -
TY - CONF
AB - A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of G+e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP -complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles A and a pseudosegment σ , it can be decided in polynomial time whether there exists a pseudocircle Φσ extending σ for which A∪{Φσ} is again an arrangement of pseudocircles.
AU - Arroyo Guevara, Alan M
AU - Klute, Fabian
AU - Parada, Irene
AU - Seidel, Raimund
AU - Vogtenhuber, Birgit
AU - Wiedera, Tilo
ID - 8732
SN - 0302-9743
T2 - Graph-Theoretic Concepts in Computer Science
TI - Inserting one edge into a simple drawing is hard
VL - 12301
ER -
TY - CONF
AB - We consider the following decision problem EMBEDk→d in computational topology (where k ≤ d are fixed positive integers): Given a finite simplicial complex K of dimension k, does there exist a (piecewise-linear) embedding of K into ℝd?
The special case EMBED1→2 is graph planarity, which is decidable in linear time, as shown by Hopcroft and Tarjan. In higher dimensions, EMBED2→3 and EMBED3→3 are known to be decidable (as well as NP-hard), and recent results of Čadek et al. in computational homotopy theory, in combination with the classical Haefliger–Weber theorem in geometric topology, imply that EMBEDk→d can be solved in polynomial time for any fixed pair (k, d) of dimensions in the so-called metastable range .
Here, by contrast, we prove that EMBEDk→d is algorithmically undecidable for almost all pairs of dimensions outside the metastable range, namely for . This almost completely resolves the decidability vs. undecidability of EMBEDk→d in higher dimensions and establishes a sharp dichotomy between polynomial-time solvability and undecidability.
Our result complements (and in a wide range of dimensions strengthens) earlier results of Matoušek, Tancer, and the second author, who showed that EMBEDk→d is undecidable for 4 ≤ k ϵ {d – 1, d}, and NP-hard for all remaining pairs (k, d) outside the metastable range and satisfying d ≥ 4.
AU - Filakovský, Marek
AU - Wagner, Uli
AU - Zhechev, Stephan Y
ID - 7806
SN - 9781611975994
T2 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
TI - Embeddability of simplicial complexes is undecidable
VL - 2020-January
ER -
TY - CONF
AB - In a straight-line embedded triangulation of a point set P in the plane, removing an inner edge and—provided the resulting quadrilateral is convex—adding the other diagonal is called an edge flip. The (edge) flip graph has all triangulations as vertices, and a pair of triangulations is adjacent if they can be obtained from each other by an edge flip. The goal of this paper is to contribute to a better understanding of the flip graph, with an emphasis on its connectivity.
For sets in general position, it is known that every triangulation allows at least edge flips (a tight bound) which gives the minimum degree of any flip graph for n points. We show that for every point set P in general position, the flip graph is at least -vertex connected. Somewhat more strongly, we show that the vertex connectivity equals the minimum degree occurring in the flip graph, i.e. the minimum number of flippable edges in any triangulation of P, provided P is large enough. Finally, we exhibit some of the geometry of the flip graph by showing that the flip graph can be covered by 1-skeletons of polytopes of dimension (products of associahedra).
A corresponding result ((n – 3)-vertex connectedness) can be shown for the bistellar flip graph of partial triangulations, i.e. the set of all triangulations of subsets of P which contain all extreme points of P. This will be treated separately in a second part.
AU - Wagner, Uli
AU - Welzl, Emo
ID - 7807
SN - 9781611975994
T2 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
TI - Connectivity of triangulation flip graphs in the plane (Part I: Edge flips)
VL - 2020-January
ER -
TY - THES
AB - This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph.
For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton.
In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars.
AU - Masárová, Zuzana
ID - 7944
KW - reconfiguration
KW - reconfiguration graph
KW - triangulations
KW - flip
KW - constrained triangulations
KW - shellability
KW - piecewise-linear balls
KW - token swapping
KW - trees
KW - coloured weighted token swapping
SN - 978-3-99078-005-3
TI - Reconfiguration problems
ER -
TY - JOUR
AB - This paper presents two algorithms. The first decides the existence of a pointed homotopy between given simplicial maps 𝑓,𝑔:𝑋→𝑌, and the second computes the group [𝛴𝑋,𝑌]∗ of pointed homotopy classes of maps from a suspension; in both cases, the target Y is assumed simply connected. More generally, these algorithms work relative to 𝐴⊆𝑋.
AU - Filakovský, Marek
AU - Vokřínek, Lukas
ID - 6563
JF - Foundations of Computational Mathematics
SN - 16153375
TI - Are two given maps homotopic? An algorithmic viewpoint
VL - 20
ER -
TY - GEN
AB - Suppose that $n\neq p^k$ and $n\neq 2p^k$ for all $k$ and all primes $p$. We prove that for any Hausdorff compactum $X$ with a free action of the symmetric group $\mathfrak S_n$ there exists an $\mathfrak S_n$-equivariant map $X \to
{\mathbb R}^n$ whose image avoids the diagonal $\{(x,x\dots,x)\in {\mathbb R}^n|x\in {\mathbb R}\}$.
Previously, the special cases of this statement for certain $X$ were usually proved using the equivartiant obstruction theory. Such calculations are difficult and may become infeasible past the first (primary) obstruction. We
take a different approach which allows us to prove the vanishing of all obstructions simultaneously. The essential step in the proof is classifying the possible degrees of $\mathfrak S_n$-equivariant maps from the boundary
$\partial\Delta^{n-1}$ of $(n-1)$-simplex to itself. Existence of equivariant maps between spaces is important for many questions arising from discrete mathematics and geometry, such as Kneser's conjecture, the Square Peg conjecture, the Splitting Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate the utility of our result applying it to one such question, a specific instance of envy-free division problem.
AU - Avvakumov, Sergey
AU - Kudrya, Sergey
ID - 8182
T2 - arXiv
TI - Vanishing of all equivariant obstructions and the mapping degree
ER -
TY - GEN
AB - Denote by ∆N the N-dimensional simplex. A map f : ∆N → Rd is an almost r-embedding if fσ1∩. . .∩fσr = ∅ whenever σ1, . . . , σr are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if r is not a prime power and d ≥ 2r + 1, then there is an almost r-embedding ∆(d+1)(r−1) → Rd. This was improved by Blagojevi´c–Frick–Ziegler using a simple construction of higher-dimensional counterexamples by taking k-fold join power of lower-dimensional ones. We improve this further (for d large compared to r): If r is not a prime power and N := (d+ 1)r−r l
d + 2 r + 1 m−2, then there is an almost r-embedding ∆N → Rd. For the r-fold van Kampen–Flores conjecture we also produce counterexamples which are stronger than previously known. Our proof is based on generalizations of the Mabillard–Wagner theorem on construction of almost r-embeddings from equivariant maps, and of the Ozaydin theorem on existence of equivariant maps.
AU - Avvakumov, Sergey
AU - Karasev, R.
AU - Skopenkov, A.
ID - 8184
T2 - arXiv
TI - Stronger counterexamples to the topological Tverberg conjecture
ER -
TY - GEN
AB - In this paper we study envy-free division problems. The classical approach to some of such problems, used by David Gale, reduces to considering continuous maps of a simplex to itself and finding sufficient conditions when this map hits the center of the simplex. The mere continuity is not sufficient for such a conclusion, the usual assumption (for example, in the Knaster--Kuratowski--Mazurkiewicz and the Gale theorem) is a certain boundary condition.
We follow Erel Segal-Halevi, Fr\'ed\'eric Meunier, and Shira Zerbib, and replace the boundary condition by another assumption, which has the economic meaning of possibility for a player to prefer an empty part in the segment
partition problem. We solve the problem positively when $n$, the number of players that divide the segment, is a prime power, and we provide counterexamples for every $n$ which is not a prime power. We also provide counterexamples relevant to a wider class of fair or envy-free partition problems when $n$ is odd and not a prime power.
AU - Avvakumov, Sergey
AU - Karasev, Roman
ID - 8185
T2 - arXiv
TI - Envy-free division using mapping degree
ER -
TY - CONF
AB - Simple drawings of graphs are those in which each pair of edges share at most one point, either a common endpoint or a proper crossing. In this paper we study the problem of extending a simple drawing D(G) of a graph G by inserting a set of edges from the complement of G into D(G) such that the result is a simple drawing. In the context of rectilinear drawings, the problem is trivial. For pseudolinear drawings, the existence of such an extension follows from Levi’s enlargement lemma. In contrast, we prove that deciding if a given set of edges can be inserted into a simple drawing is NP-complete. Moreover, we show that the maximization version of the problem is APX-hard. We also present a polynomial-time algorithm for deciding whether one edge uv can be inserted into D(G) when {u,v} is a dominating set for the graph G.
AU - Arroyo Guevara, Alan M
AU - Derka, Martin
AU - Parada, Irene
ID - 7230
SN - 03029743
T2 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
TI - Extending simple drawings
VL - 11904
ER -
TY - CONF
AB - The genus g(G) of a graph G is the minimum g such that G has an embedding on the orientable surface M_g of genus g. A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z_2-genus of a graph G, denoted by g_0(G), is the minimum g such that G has an independently even drawing on M_g. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Stefankovic proved that the Z_2-genus of a graph is additive over 2-connected blocks as well, and asked whether this result can be extended to so-called 2-amalgamations, as an analogue of results by Decker, Glover, Huneke, and Stahl for the genus. We give the following partial answer. If G=G_1 cup G_2, G_1 and G_2 intersect in two vertices u and v, and G-u-v has k connected components (among which we count the edge uv if present), then |g_0(G)-(g_0(G_1)+g_0(G_2))|<=k+1. For complete bipartite graphs K_{m,n}, with n >= m >= 3, we prove that g_0(K_{m,n})/g(K_{m,n})=1-O(1/n). Similar results are proved also for the Euler Z_2-genus. We express the Z_2-genus of a graph using the minimum rank of partial symmetric matrices over Z_2; a problem that might be of independent interest.
AU - Fulek, Radoslav
AU - Kyncl, Jan
ID - 7401
SN - 1868-8969
T2 - 35th International Symposium on Computational Geometry (SoCG 2019)
TI - Z_2-Genus of graphs and minimum rank of partial symmetric matrices
VL - 129
ER -
TY - GEN
AB - The input to the token swapping problem is a graph with vertices v1, v2, . . . , vn, and n tokens with labels 1,2, . . . , n, one on each vertex. The goal is to get token i to vertex vi for all i= 1, . . . , n using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge.Token swapping on a tree, also known as “sorting with a transposition tree,” is not known to be in P nor NP-complete. We present some partial results:
1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a “happy leaf”), disproving a conjecture of Vaughan.
2. Any algorithm that fixes happy leaves—as all known approximation algorithms for the problem do—has approximation factor at least 4/3. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2.
3. A generalized problem—weighted coloured token swapping—is NP-complete on trees, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.
AU - Biniaz, Ahmad
AU - Jain, Kshitij
AU - Lubiw, Anna
AU - Masárová, Zuzana
AU - Miltzow, Tillmann
AU - Mondal, Debajyoti
AU - Naredla, Anurag Murty
AU - Tkadlec, Josef
AU - Turcotte, Alexi
ID - 7950
T2 - arXiv
TI - Token swapping on trees
ER -
TY - JOUR
AB - The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle graphs, where the input consists of a graph G and a partial representation R′ giving some predrawn chords that represent an induced subgraph of G. The question is whether one can extend R′ to a representation R of the entire graph G, that is, whether one can draw the remaining chords into a partially predrawn representation to obtain a representation of G. Our main result is an O(n3) time algorithm for partial representation extension of circle graphs, where n is the number of vertices. To show this, we describe the structure of all representations of a circle graph using split decomposition. This can be of independent interest.
AU - Chaplick, Steven
AU - Fulek, Radoslav
AU - Klavík, Pavel
ID - 5790
IS - 4
JF - Journal of Graph Theory
SN - 03649024
TI - Extending partial representations of circle graphs
VL - 91
ER -
TY - JOUR
AB - A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an odd number of times. It is also shown that the maximum number of edges of a quasi-thrackle on n vertices is [Formula presented](n−1), and that this bound is best possible for infinitely many values of n.
AU - Fulek, Radoslav
AU - Pach, János
ID - 5857
IS - 4
JF - Discrete Applied Mathematics
SN - 0166218X
TI - Thrackles: An improved upper bound
VL - 259
ER -
TY - JOUR
AB - Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomial-time algorithm (with 𝑂(𝑛8) being a crude bound on the run-time) to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of 𝑂(𝑛7) on the length of the flip sequence. Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture.
AU - Lubiw, Anna
AU - Masárová, Zuzana
AU - Wagner, Uli
ID - 5986
IS - 4
JF - Discrete & Computational Geometry
SN - 0179-5376
TI - A proof of the orbit conjecture for flipping edge-labelled triangulations
VL - 61
ER -
TY - CONF
AB - Motivated by fixed-parameter tractable (FPT) problems in computational topology, we consider the treewidth tw(M) of a compact, connected 3-manifold M, defined to be the minimum treewidth of the face pairing graph of any triangulation T of M. In this setting the relationship between the topology of a 3-manifold and its treewidth is of particular interest. First, as a corollary of work of Jaco and Rubinstein, we prove that for any closed, orientable 3-manifold M the treewidth tw(M) is at most 4g(M)-2, where g(M) denotes Heegaard genus of M. In combination with our earlier work with Wagner, this yields that for non-Haken manifolds the Heegaard genus and the treewidth are within a constant factor. Second, we characterize all 3-manifolds of treewidth one: These are precisely the lens spaces and a single other Seifert fibered space. Furthermore, we show that all remaining orientable Seifert fibered spaces over the 2-sphere or a non-orientable surface have treewidth two. In particular, for every spherical 3-manifold we exhibit a triangulation of treewidth at most two. Our results further validate the parameter of treewidth (and other related parameters such as cutwidth or congestion) to be useful for topological computing, and also shed more light on the scope of existing FPT-algorithms in the field.
AU - Huszár, Kristóf
AU - Spreer, Jonathan
ID - 6556
KW - computational 3-manifold topology
KW - fixed-parameter tractability
KW - layered triangulations
KW - structural graph theory
KW - treewidth
KW - cutwidth
KW - Heegaard genus
SN - 1868-8969
T2 - 35th International Symposium on Computational Geometry
TI - 3-manifold triangulations with small treewidth
VL - 129
ER -
TY - JOUR
AB - The crossing number of a graph G is the least number of crossings over all possible drawings of G. We present a structural characterization of graphs with crossing number one.
AU - Silva, André
AU - Arroyo Guevara, Alan M
AU - Richter, Bruce
AU - Lee, Orlando
ID - 6638
IS - 11
JF - Discrete Mathematics
SN - 0012-365X
TI - Graphs with at most one crossing
VL - 342
ER -
TY - CONF
AB - The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2.
AU - Fulek, Radoslav
AU - Gärtner, Bernd
AU - Kupavskii, Andrey
AU - Valtr, Pavel
AU - Wagner, Uli
ID - 6647
SN - 1868-8969
T2 - 35th International Symposium on Computational Geometry
TI - The crossing Tverberg theorem
VL - 129
ER -
TY - THES
AB - The first part of the thesis considers the computational aspects of the homotopy groups πd(X) of a topological space X. It is well known that there is no algorithm to decide whether the fundamental group π1(X) of a given finite simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex X that is simply connected (i.e., with π1(X) trivial), compute the higher homotopy group πd(X) for any given d ≥ 2.
However, these algorithms come with a caveat: They compute the isomorphism type of πd(X), d ≥ 2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of πd(X). We present an algorithm that, given a simply connected space X, computes πd(X) and represents its elements as simplicial maps from suitable triangulations of the d-sphere Sd to X. For fixed d, the algorithm runs in time exponential in size(X), the number of simplices of X. Moreover, we prove that this is optimal: For every fixed d ≥ 2,
we construct a family of simply connected spaces X such that for any simplicial map representing a generator of πd(X), the size of the triangulation of S d on which the map is defined, is exponential in size(X).
In the second part of the thesis, we prove that the following question is algorithmically undecidable for d < ⌊3(k+1)/2⌋, k ≥ 5 and (k, d) ̸= (5, 7), which covers essentially everything outside the meta-stable range: Given a finite simplicial complex K of dimension k, decide whether there exists a piecewise-linear (i.e., linear on an arbitrarily fine subdivision of K) embedding f : K ↪→ Rd of K into a d-dimensional Euclidean space.
AU - Zhechev, Stephan Y
ID - 6681
SN - 2663-337X
TI - Algorithmic aspects of homotopy theory and embeddability
ER -
TY - JOUR
AB - We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding ϕ : G → M of a graph G into a 2-manifold M maps the vertices in V(G) to distinct points and the edges in E(G) to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to the same point or overlapping arcs due to data compression or low resolution. This raises the computational problem of deciding whether a given map ϕ : G → M comes from an embedding. A map ϕ : G → M is a weak embedding if it can be perturbed into an embedding ψ ϵ : G → M with ‖ ϕ − ψ ϵ ‖ < ϵ for every ϵ > 0, where ‖.‖ is the unform norm.
A polynomial-time algorithm for recognizing weak embeddings has recently been found by Fulek and Kynčl. It reduces the problem to solving a system of linear equations over Z2. It runs in O(n2ω)≤ O(n4.75) time, where ω ∈ [2,2.373) is the matrix multiplication exponent and n is the number of vertices and edges of G. We improve the running time to O(n log n). Our algorithm is also conceptually simpler: We perform a sequence of local operations that gradually “untangles” the image ϕ(G) into an embedding ψ(G) or reports that ϕ is not a weak embedding. It combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.
AU - Akitaya, Hugo
AU - Fulek, Radoslav
AU - Tóth, Csaba
ID - 6982
IS - 4
JF - ACM Transactions on Algorithms
TI - Recognizing weak embeddings of graphs
VL - 15
ER -
TY - JOUR
AB - We find a graph of genus 5 and its drawing on the orientable surface of genus 4 with every pair of independent edges crossing an even number of times. This shows that the strong Hanani–Tutte theorem cannot be extended to the orientable surface of genus 4. As a base step in the construction we use a counterexample to an extension of the unified Hanani–Tutte theorem on the torus.
AU - Fulek, Radoslav
AU - Kynčl, Jan
ID - 7034
IS - 6
JF - Combinatorica
SN - 0209-9683
TI - Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4
VL - 39
ER -
TY - JOUR
AB - In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic problems; in particular, there is a variety of problems in computational 3-manifold topology - some of them known to be computationally hard in general - that become solvable in polynomial time as soon as the dual graph of the input triangulation has bounded treewidth.
In view of these algorithmic results, it is natural to ask whether every 3-manifold admits a triangulation of bounded treewidth. We show that this is not the case, i.e., that there exists an infinite family of closed 3-manifolds not admitting triangulations of bounded pathwidth or treewidth (the latter implies the former, but we present two separate proofs).
We derive these results from work of Agol, of Scharlemann and Thompson, and of Scharlemann, Schultens and Saito by exhibiting explicit connections between the topology of a 3-manifold M on the one hand and width-type parameters of the dual graphs of triangulations of M on the other hand, answering a question that had been raised repeatedly by researchers in computational 3-manifold topology. In particular, we show that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth (resp. pathwidth) k then the Heegaard genus of M is at most 18(k+1) (resp. 4(3k+1)).
AU - Huszár, Kristóf
AU - Spreer, Jonathan
AU - Wagner, Uli
ID - 7093
IS - 2
JF - Journal of Computational Geometry
SN - 1920-180X
TI - On the treewidth of triangulated 3-manifolds
VL - 10
ER -
TY - JOUR
AB - We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 and k ≥ 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to contractible pure d-dimensional complexes. Another simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-shellable.
AU - Goaoc, Xavier
AU - Patak, Pavel
AU - Patakova, Zuzana
AU - Tancer, Martin
AU - Wagner, Uli
ID - 7108
IS - 3
JF - Journal of the ACM
SN - 0004-5411
TI - Shellability is NP-complete
VL - 66
ER -
TY - CONF
AB - In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic problems; in particular, there is a variety of problems in computational 3-manifold topology - some of them known to be computationally hard in general - that become solvable in polynomial time as soon as the dual graph of the input triangulation has bounded treewidth. In view of these algorithmic results, it is natural to ask whether every 3-manifold admits a triangulation of bounded treewidth. We show that this is not the case, i.e., that there exists an infinite family of closed 3-manifolds not admitting triangulations of bounded pathwidth or treewidth (the latter implies the former, but we present two separate proofs). We derive these results from work of Agol and of Scharlemann and Thompson, by exhibiting explicit connections between the topology of a 3-manifold M on the one hand and width-type parameters of the dual graphs of triangulations of M on the other hand, answering a question that had been raised repeatedly by researchers in computational 3-manifold topology. In particular, we show that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth (resp. pathwidth) k then the Heegaard genus of M is at most 48(k+1) (resp. 4(3k+1)).
AU - Huszár, Kristóf
AU - Spreer, Jonathan
AU - Wagner, Uli
ID - 285
SN - 18688969
TI - On the treewidth of triangulated 3-manifolds
VL - 99
ER -
TY - CONF
AB - We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 and k ≥ 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to contractible pure d-dimensional complexes.
AU - Goaoc, Xavier
AU - Paták, Pavel
AU - Patakova, Zuzana
AU - Tancer, Martin
AU - Wagner, Uli
ID - 184
TI - Shellability is NP-complete
VL - 99
ER -
TY - CONF
AB - We resolve in the affirmative conjectures of A. Skopenkov and Repovš (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our proof of this result is constructive and almost immediately implies an efficient algorithm for testing whether a given piecewise linear map of a graph in a surface is approximable by an embedding. More precisely, an instance of this problem consists of (i) a graph G whose vertices are partitioned into clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a region R of a 2-dimensional compact surface M given as the union of a set of pairwise disjoint discs corresponding to the clusters and a set of pairwise disjoint "pipes" corresponding to the bundles, connecting certain pairs of these discs. We are to decide whether G can be embedded inside M so that the vertices in every cluster are drawn in the corresponding disc, the edges in every bundle pass only through its corresponding pipe, and every edge crosses the boundary of each disc at most once.
AU - Fulek, Radoslav
AU - Kynčl, Jan
ID - 185
SN - 978-3-95977-066-8
TI - Hanani-Tutte for approximating maps of graphs
VL - 99
ER -
TY - CONF
AB - A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The ℤ2-genus of a graph G is the minimum g such that G has an independently even drawing on the orientable surface of genus g. An unpublished result by Robertson and Seymour implies that for every t, every graph of sufficiently large genus contains as a minor a projective t × t grid or one of the following so-called t-Kuratowski graphs: K3, t, or t copies of K5 or K3,3 sharing at most 2 common vertices. We show that the ℤ2-genus of graphs in these families is unbounded in t; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its ℤ2-genus, solving a problem posed by Schaefer and Štefankovič, and giving an approximate version of the Hanani-Tutte theorem on orientable surfaces.
AU - Fulek, Radoslav
AU - Kynčl, Jan
ID - 186
TI - The ℤ2-Genus of Kuratowski minors
VL - 99
ER -
TY - JOUR
AB - We give a detailed and easily accessible proof of Gromov’s Topological Overlap Theorem. Let X be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension d. Informally, the theorem states that if X has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of X) then X has the following topological overlap property: for every continuous map (Formula presented.) there exists a point (Formula presented.) that is contained in the images of a positive fraction (Formula presented.) of the d-cells of X. More generally, the conclusion holds if (Formula presented.) is replaced by any d-dimensional piecewise-linear manifold M, with a constant (Formula presented.) that depends only on d and on the expansion properties of X, but not on M.
AU - Dotterrer, Dominic
AU - Kaufman, Tali
AU - Wagner, Uli
ID - 742
IS - 1
JF - Geometriae Dedicata
TI - On expansion and topological overlap
VL - 195
ER -
TY - CONF
AB - Due to data compression or low resolution, nearby vertices and edges of a graph drawing may be bundled to a common node or arc. We model such a “compromised” drawing by a piecewise linear map φ:G → ℝ. We wish to perturb φ by an arbitrarily small ε>0 into a proper drawing (in which the vertices are distinct points, any two edges intersect in finitely many points, and no three edges have a common interior point) that minimizes the number of crossings. An ε-perturbation, for every ε>0, is given by a piecewise linear map (Formula Presented), where with ||·|| is the uniform norm (i.e., sup norm). We present a polynomial-time solution for this optimization problem when G is a cycle and the map φ has no spurs (i.e., no two adjacent edges are mapped to overlapping arcs). We also show that the problem becomes NP-complete (i) when G is an arbitrary graph and φ has no spurs, and (ii) when φ may have spurs and G is a cycle or a union of disjoint paths.
AU - Fulek, Radoslav
AU - Tóth, Csaba D.
ID - 5791
SN - 9783030044138
TI - Crossing minimization in perturbed drawings
VL - 11282
ER -
TY - JOUR
AB - In this paper we present a reliable method to verify the existence of loops along the uncertain trajectory of a robot, based on proprioceptive measurements only, within a bounded-error context. The loop closure detection is one of the key points in simultaneous localization and mapping (SLAM) methods, especially in homogeneous environments with difficult scenes recognitions. The proposed approach is generic and could be coupled with conventional SLAM algorithms to reliably reduce their computing burden, thus improving the localization and mapping processes in the most challenging environments such as unexplored underwater extents. To prove that a robot performed a loop whatever the uncertainties in its evolution, we employ the notion of topological degree that originates in the field of differential topology. We show that a verification tool based on the topological degree is an optimal method for proving robot loops. This is demonstrated both on datasets from real missions involving autonomous underwater vehicles and by a mathematical discussion.
AU - Rohou, Simon
AU - Franek, Peter
AU - Aubry, Clément
AU - Jaulin, Luc
ID - 5960
IS - 12
JF - The International Journal of Robotics Research
SN - 0278-3649
TI - Proving the existence of loops in robot trajectories
VL - 37
ER -
TY - JOUR
AB - We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle.
AU - Akopyan, Arseniy
AU - Avvakumov, Sergey
ID - 6355
JF - Forum of Mathematics, Sigma
SN - 2050-5094
TI - Any cyclic quadrilateral can be inscribed in any closed convex smooth curve
VL - 6
ER -
TY - JOUR
AB - A central problem of algebraic topology is to understand the homotopy groups 𝜋𝑑(𝑋) of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group 𝜋1(𝑋) of a given finite simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex X that is simply connected (i.e., with 𝜋1(𝑋) trivial), compute the higher homotopy group 𝜋𝑑(𝑋) for any given 𝑑≥2 . However, these algorithms come with a caveat: They compute the isomorphism type of 𝜋𝑑(𝑋) , 𝑑≥2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of 𝜋𝑑(𝑋) . Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere 𝑆𝑑 to X has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a simply connected space X, computes 𝜋𝑑(𝑋) and represents its elements as simplicial maps from a suitable triangulation of the d-sphere 𝑆𝑑 to X. For fixed d, the algorithm runs in time exponential in size(𝑋) , the number of simplices of X. Moreover, we prove that this is optimal: For every fixed 𝑑≥2 , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of 𝜋𝑑(𝑋) , the size of the triangulation of 𝑆𝑑 on which the map is defined, is exponential in size(𝑋) .
AU - Filakovský, Marek
AU - Franek, Peter
AU - Wagner, Uli
AU - Zhechev, Stephan Y
ID - 6774
IS - 3-4
JF - Journal of Applied and Computational Topology
SN - 2367-1726
TI - Computing simplicial representatives of homotopy group elements
VL - 2
ER -
TY - JOUR
AB - We show that the following algorithmic problem is decidable: given a 2-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in R3? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold X into the 3-sphere S3. The main step, which allows us to simplify X and recurse, is in proving that if X can be embedded in S3, then there is also an embedding in which X has a short meridian, that is, an essential curve in the boundary of X bounding a disk in S3 \ X with length bounded by a computable function of the number of tetrahedra of X.
AU - Matoušek, Jiří
AU - Sedgwick, Eric
AU - Tancer, Martin
AU - Wagner, Uli
ID - 425
IS - 1
JF - Journal of the ACM
TI - Embeddability in the 3-Sphere is decidable
VL - 65
ER -
TY - CONF
AB - A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an odd number of times. It is also shown that the maximum number of edges of a quasi-thrackle on n vertices is 3/2(n-1), and that this bound is best possible for infinitely many values of n.
AU - Fulek, Radoslav
AU - Pach, János
ID - 433
TI - Thrackles: An improved upper bound
VL - 10692
ER -
TY - CONF
AB - We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding ' : G ! M of a graph G into a 2manifold M maps the vertices in V (G) to distinct points and the edges in E(G) to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to a common node or arc, due to data compression or low resolution. This raises the computational problem of deciding whether a given map ' : G ! M comes from an embedding. A map ' : G ! M is a weak embedding if it can be perturbed into an embedding ψ: G ! M with k' "k < " for every " > 0. A polynomial-time algorithm for recognizing weak embeddings was recently found by Fulek and Kyncl [14], which reduces to solving a system of linear equations over Z2. It runs in O(n2!) O(n4:75) time, where 2:373 is the matrix multiplication exponent and n is the number of vertices and edges of G. We improve the running time to O(n log n). Our algorithm is also conceptually simpler than [14]: We perform a sequence of local operations that gradually "untangles" the image '(G) into an embedding (G), or reports that ' is not a weak embedding. It generalizes a recent technique developed for the case that G is a cycle and the embedding is a simple polygon [1], and combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.
AU - Akitaya, Hugo
AU - Fulek, Radoslav
AU - Tóth, Csaba
ID - 309
TI - Recognizing weak embeddings of graphs
ER -
TY - JOUR
AB - Let P be a finite point set in the plane. A cordinary triangle in P is a subset of P consisting of three non-collinear points such that each of the three lines determined by the three points contains at most c points of P . Motivated by a question of Erdös, and answering a question of de Zeeuw, we prove that there exists a constant c > 0such that P contains a c-ordinary triangle, provided that P is not contained in the union of two lines. Furthermore, the number of c-ordinary triangles in P is Ω(| P |).
AU - Fulek, Radoslav
AU - Mojarrad, Hossein
AU - Naszódi, Márton
AU - Solymosi, József
AU - Stich, Sebastian
AU - Szedlák, May
ID - 793
JF - Computational Geometry: Theory and Applications
SN - 09257721
TI - On the existence of ordinary triangles
VL - 66
ER -
TY - JOUR
AB - We show that c-planarity is solvable in quadratic time for flat clustered graphs with three clusters if the combinatorial embedding of the underlying graph is fixed. In simpler graph-theoretical terms our result can be viewed as follows. Given a graph G with the vertex set partitioned into three parts embedded on a 2-sphere, our algorithm decides if we can augment G by adding edges without creating an edge-crossing so that in the resulting spherical graph the vertices of each part induce a connected sub-graph. We proceed by a reduction to the problem of testing the existence of a perfect matching in planar bipartite graphs. We formulate our result in a slightly more general setting of cyclic clustered graphs, i.e., the simple graph obtained by contracting each cluster, where we disregard loops and multi-edges, is a cycle.
AU - Fulek, Radoslav
ID - 794
JF - Computational Geometry: Theory and Applications
TI - C-planarity of embedded cyclic c-graphs
VL - 66
ER -
TY - JOUR
AB - We introduce a common generalization of the strong Hanani–Tutte theorem and the weak Hanani–Tutte theorem: if a graph G has a drawing D in the plane where every pair of independent edges crosses an even number of times, then G has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in D. The theorem is implicit in the proof of the strong Hanani–Tutte theorem by Pelsmajer, Schaefer and Štefankovič. We give a new, somewhat simpler proof.
AU - Fulek, Radoslav
AU - Kynčl, Jan
AU - Pálvölgyi, Dömötör
ID - 795
IS - 3
JF - Electronic Journal of Combinatorics
SN - 10778926
TI - Unified Hanani Tutte theorem
VL - 24
ER -
TY - JOUR
AB - We investigate the complexity of finding an embedded non-orientable surface of Euler genus g in a triangulated 3-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into 3-manifolds. We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.
AU - Burton, Benjamin
AU - De Mesmay, Arnaud N
AU - Wagner, Uli
ID - 534
IS - 4
JF - Discrete & Computational Geometry
SN - 01795376
TI - Finding non-orientable surfaces in 3-Manifolds
VL - 58
ER -
TY - JOUR
AB - We study robust properties of zero sets of continuous maps f: X → ℝn. Formally, we analyze the family Z< r(f) := (g-1(0): ||g - f|| < r) of all zero sets of all continuous maps g closer to f than r in the max-norm. All of these sets are outside A := (x: |f(x)| ≥ r) and we claim that Z< r(f) is fully determined by A and an element of a certain cohomotopy group which (by a recent result) is computable whenever the dimension of X is at most 2n - 3. By considering all r > 0 simultaneously, the pointed cohomotopy groups form a persistence module-a structure leading to persistence diagrams as in the case of persistent homology or well groups. Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams. Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C).
AU - Franek, Peter
AU - Krcál, Marek
ID - 568
IS - 2
JF - Homology, Homotopy and Applications
SN - 15320073
TI - Persistence of zero sets
VL - 19
ER -
TY - JOUR
AB - The fact that the complete graph K5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph Kn embeds in a closed surface M (other than the Klein bottle) if and only if (n−3)(n−4) ≤ 6b1(M), where b1(M) is the first Z2-Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of Kn+1) embeds in R2k if and only if n ≤ 2k + 1. Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k − 1)-connected 2k-manifold with kth Z2-Betti number bk only if the following generalized Heawood inequality holds: (k+1 n−k−1) ≤ (k+1 2k+1)bk. This is a common generalization of the case of graphs on surfaces as well as the van Kampen–Flores theorem. In the spirit of Kühnel’s conjecture, we prove that if the k-skeleton of the n-simplex embeds in a compact 2k-manifold with kth Z2-Betti number bk, then n ≤ 2bk(k 2k+2)+2k+4. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k−1)-connected. Our results generalize to maps without q-covered points, in the spirit of Tverberg’s theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.
AU - Goaoc, Xavier
AU - Mabillard, Isaac
AU - Paták, Pavel
AU - Patakova, Zuzana
AU - Tancer, Martin
AU - Wagner, Uli
ID - 610
IS - 2
JF - Israel Journal of Mathematics
TI - On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result
VL - 222
ER -
TY - CONF
AB - A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing, whether a drawing of a planar graph G in the plane is approximable by an embedding, can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class, i.e., the rotation system (or equivalently the faces) of the embedding of G and the choice of outer face are fixed. In other words, we show that c-planarity with embedded pipes is tractable for graphs with fixed embeddings. To the best of our knowledge an analogous result was previously known essentially only when G is a cycle.
AU - Fulek, Radoslav
ID - 6517
TI - Embedding graphs into embedded graphs
VL - 92
ER -
TY - CONF
AB - Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomial-time algorithm to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of O(n7) on the length of the flip sequence. Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture.
AU - Lubiw, Anna
AU - Masárová, Zuzana
AU - Wagner, Uli
ID - 683
TI - A proof of the orbit conjecture for flipping edge labelled triangulations
VL - 77
ER -
TY - CONF
AB - We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. Examples are the Kullback - Leibler divergence, which is commonly used for comparing text and images, and the Itakura - Saito divergence, popular for speech and sound. In particular, we prove that appropriately generalized čech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized čech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory.
AU - Edelsbrunner, Herbert
AU - Wagner, Hubert
ID - 688
SN - 18688969
TI - Topological data analysis with Bregman divergences
VL - 77
ER -
TY - JOUR
AB - A d-dimensional simplex S is called a k-reptile (or a k-reptile simplex) if it can be tiled by k simplices with disjoint interiors that are all mutually congruent and similar to S. For d = 2, triangular k-reptiles exist for all k of the form a^2, 3a^2 or a^2+b^2 and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only k-reptile simplices that are known for d ≥ 3, have k = m^d, where m is a positive integer. We substantially simplify the proof by Matoušek and the second author that for d = 3, k-reptile tetrahedra can exist only for k = m^3. We then prove a weaker analogue of this result for d = 4 by showing that four-dimensional k-reptile simplices can exist only for k = m^2.
AU - Kynčl, Jan
AU - Patakova, Zuzana
ID - 701
IS - 3
JF - The Electronic Journal of Combinatorics
SN - 10778926
TI - On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4
VL - 24
ER -
TY - JOUR
AB - Let X and Y be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group G. Assuming that Y is d-connected and dimX≤2d, for some d≥1, we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps |X|→|Y|; the existence of such a map can be decided even for dimX≤2d+1. This yields the first algorithm for deciding topological embeddability of a k-dimensional finite simplicial complex into Rn under the condition k≤23n−1. More generally, we present an algorithm that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions. This result is new even in the non-equivariant situation.
AU - Čadek, Martin
AU - Krcál, Marek
AU - Vokřínek, Lukáš
ID - 1073
IS - 4
JF - Discrete & Computational Geometry
SN - 01795376
TI - Algorithmic solvability of the lifting extension problem
VL - 54
ER -
TY - JOUR
AB - A drawing of a graph G is radial if the vertices of G are placed on concentric circles C 1 , . . . , C k with common center c , and edges are drawn radially : every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. We show that a graph G is radial planar if G has a radial drawing in which every two edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Toth.
AU - Fulek, Radoslav
AU - Pelsmajer, Michael
AU - Schaefer, Marcus
ID - 1113
IS - 1
JF - Journal of Graph Algorithms and Applications
TI - Hanani-Tutte for radial planarity
VL - 21
ER -
TY - CHAP
AB - We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b, d) such that the following holds. If F is a finite family of subsets of Rd such that βi(∩G)≤b for any G⊊F and every 0 ≤ i ≤ [d/2]-1 then F has Helly number at most h(b, d). Here βi denotes the reduced Z2-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these [d/2] first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map C*(K)→C*(Rd).
AU - Goaoc, Xavier
AU - Paták, Pavel
AU - Patakova, Zuzana
AU - Tancer, Martin
AU - Wagner, Uli
ED - Loebl, Martin
ED - Nešetřil, Jaroslav
ED - Thomas, Robin
ID - 424
SN - 978-331944479-6
T2 - A Journey through Discrete Mathematics: A Tribute to Jiri Matousek
TI - Bounding helly numbers via betti numbers
ER -
TY - JOUR
AB - We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial–Meshulam model Xk(n, p) of random k-dimensional simplicial complexes on n vertices. We show that for p = Ω(logn/n), the eigenvalues of each of the matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of (k - 2)-dimensional faces. Garland’s result concerns the Laplacian; we develop an analogous result for the adjacency matrix. The same arguments apply to other models of random complexes which allow for dependencies between the choices of k-dimensional simplices. In the second part of the paper, we apply this to the question of possible higher-dimensional analogues of the discrete Cheeger inequality, which in the classical case of graphs relates the eigenvalues of a graph and its edge expansion. It is very natural to ask whether this generalizes to higher dimensions and, in particular, whether the eigenvalues of the higher-dimensional Laplacian capture the notion of coboundary expansion—a higher-dimensional generalization of edge expansion that arose in recent work of Linial and Meshulam and of Gromov; this question was raised, for instance, by Dotterrer and Kahle. We show that this most straightforward version of a higher-dimensional discrete Cheeger inequality fails, in quite a strong way: For every k ≥ 2 and n ∈ N, there is a k-dimensional complex Yn k on n vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised k-dimensional Laplacian lie in the interval [1−O(1/√1), 1+0(1/√1]) but whose coboundary expansion is bounded from above by O(log n/n) and so tends to zero as n → ∞; moreover, Yn k can be taken to have vanishing integer homology in dimension less than k.
AU - Gundert, Anna
AU - Wagner, Uli
ID - 1282
IS - 2
JF - Israel Journal of Mathematics
TI - On eigenvalues of random complexes
VL - 216
ER -
TY - CONF
AB - A drawing in the plane (ℝ2) of a graph G = (V,E) equipped with a function γ : V → ℕ is x-bounded if (i) x(u) < x(v) whenever γ(u) < γ(v) and (ii) γ(u) ≤ γ(w) ≤ γ(v), where uv ∈ E and γ(u) ≤ γ(v), whenever x(w) ∈ x(uv), where x(.) denotes the projection to the xaxis.We prove a characterization of isotopy classes of embeddings of connected graphs equipped with γ in the plane containing an x-bounded embedding.Then we present an efficient algorithm, which relies on our result, for testing the existence of an x-bounded embedding if the given graph is a forest.This partially answers a question raised recently by Angelini et al.and Chang et al., and proves that c-planarity testing of flat clustered graphs with three clusters is tractable when the underlying abstract graph is a forest.
AU - Fulek, Radoslav
ID - 1348
TI - Bounded embeddings of graphs in the plane
VL - 9843
ER -
TY - CONF
AB - We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let X be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension d. Informally, the theorem states that if X has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of X) then X has the following topological overlap property: for every continuous map X → ℝd there exists a point p ∈ ℝd whose preimage intersects a positive fraction μ > 0 of the d-cells of X. More generally, the conclusion holds if ℝd is replaced by any d-dimensional piecewise-linear (PL) manifold M, with a constant μ that depends only on d and on the expansion properties of X, but not on M.
AU - Dotterrer, Dominic
AU - Kaufman, Tali
AU - Wagner, Uli
ID - 1378
TI - On expansion and topological overlap
VL - 51
ER -
TY - CONF
AB - We investigate the complexity of finding an embedded non-orientable surface of Euler genus g in a triangulated 3-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into 3-manifolds. We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.
AU - Burton, Benjamin
AU - De Mesmay, Arnaud N
AU - Wagner, Uli
ID - 1379
TI - Finding non-orientable surfaces in 3-manifolds
VL - 51
ER -
TY - CONF
AB - Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into double-struck Rd without higher-multiplicity intersections. We focus on conditions for the existence of almost r-embeddings, i.e., maps f : K → double-struck Rd such that f(σ1) ∩ ⋯ ∩ f(σr) = ∅ whenever σ1, ..., σr are pairwise disjoint simplices of K. Generalizing the classical Haefliger-Weber embeddability criterion, we show that a well-known necessary deleted product condition for the existence of almost r-embeddings is sufficient in a suitable r-metastable range of dimensions: If rd ≥ (r + 1) dim K + 3, then there exists an almost r-embedding K → double-struck Rd if and only if there exists an equivariant map (K)Δ r → Sr Sd(r-1)-1, where (K)Δ r is the deleted r-fold product of K, the target Sd(r-1)-1 is the sphere of dimension d(r - 1) - 1, and Sr is the symmetric group. This significantly extends one of the main results of our previous paper (which treated the special case where d = rk and dim K = (r - 1)k for some k ≥ 3), and settles an open question raised there.
AU - Mabillard, Isaac
AU - Wagner, Uli
ID - 1381
TI - Eliminating higher-multiplicity intersections, II. The deleted product criterion in the r-metastable range
VL - 51
ER -
TY - JOUR
AB - The concept of well group in a special but important case captures homological properties of the zero set of a continuous map (Formula presented.) on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within (Formula presented.) distance r from f for a given (Formula presented.). The main drawback of the approach is that the computability of well groups was shown only when (Formula presented.) or (Formula presented.). Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of (Formula presented.) by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and (Formula presented.), our approximation of the (Formula presented.)th well group is exact. For the second part, we find examples of maps (Formula presented.) with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.
AU - Franek, Peter
AU - Krcál, Marek
ID - 1408
IS - 1
JF - Discrete & Computational Geometry
TI - On computability and triviality of well groups
VL - 56
ER -
TY - JOUR
AB - We consider two systems (α1, …, αm) and (β1, …,βn) of simple curves drawn on a compact two-dimensional surface M with boundary. Each αi and each βj is either an arc meeting the boundary of M at its two endpoints, or a closed curve. The αi are pairwise disjoint except for possibly sharing endpoints, and similarly for the βj. We want to “untangle” the βj from the ai by a self-homeomorphism of M; more precisely, we seek a homeomorphism φ:M→M fixing the boundary of M pointwise such that the total number of crossings of the ai with the φ(βj) is as small as possible. This problem is motivated by an application in the algorithmic theory of embeddings and 3-manifolds. We prove that if M is planar, i.e., a sphere with h ≥ 0 boundary components (“holes”), then O(mn) crossings can be achieved (independently of h), which is asymptotically tight, as an easy lower bound shows. In general, for an arbitrary (orientable or nonorientable) surface M with h holes and of (orientable or nonorientable) genus g ≥ 0, we obtain an O((m + n)4) upper bound, again independent of h and g. The proofs rely, among other things, on a result concerning simultaneous planar drawings of graphs by Erten and Kobourov.
AU - Matoušek, Jiří
AU - Sedgwick, Eric
AU - Tancer, Martin
AU - Wagner, Uli
ID - 1411
IS - 1
JF - Israel Journal of Mathematics
TI - Untangling two systems of noncrossing curves
VL - 212
ER -
TY - JOUR
AB - We classify smooth Brunnian (i.e., unknotted on both components) embeddings (S2 × S1) ⊔ S3 → ℝ6. Any Brunnian embedding (S2 × S1) ⊔ S3 → ℝ6 is isotopic to an explicitly constructed embedding fk,m,n for some integers k, m, n such that m ≡ n (mod 2). Two embeddings fk,m,n and fk′ ,m′,n′ are isotopic if and only if k = k′, m ≡ m′ (mod 2k) and n ≡ n′ (mod 2k). We use Haefliger’s classification of embeddings S3 ⊔ S3 → ℝ6 in our proof. The relation between the embeddings (S2 × S1) ⊔ S3 → ℝ6 and S3 ⊔ S3 → ℝ6 is not trivial, however. For example, we show that there exist embeddings f: (S2 ×S1) ⊔ S3 → ℝ6 and g, g′ : S3 ⊔ S3 → ℝ6 such that the componentwise embedded connected sum f # g is isotopic to f # g′ but g is not isotopic to g′.
AU - Avvakumov, Serhii
ID - 1522
IS - 1
JF - Moscow Mathematical Journal
TI - The classification of certain linked 3-manifolds in 6-space
VL - 16
ER -
TY - JOUR
AB - For random graphs, the containment problem considers the probability that a binomial random graph G(n, p) contains a given graph as a substructure. When asking for the graph as a topological minor, i.e., for a copy of a subdivision of the given graph, it is well known that the (sharp) threshold is at p = 1/n. We consider a natural analogue of this question for higher-dimensional random complexes Xk(n, p), first studied by Cohen, Costa, Farber and Kappeler for k = 2. Improving previous results, we show that p = Θ(1/ √n) is the (coarse) threshold for containing a subdivision of any fixed complete 2-complex. For higher dimensions k > 2, we get that p = O(n−1/k) is an upper bound for the threshold probability of containing a subdivision of a fixed k-dimensional complex.
AU - Gundert, Anna
AU - Wagner, Uli
ID - 1523
IS - 4
JF - Proceedings of the American Mathematical Society
TI - On topological minors in random simplicial complexes
VL - 144
ER -
TY - CONF
AB - A drawing of a graph G is radial if the vertices of G are placed on concentric circles C1, … , Ck with common center c, and edges are drawn radially: every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. A pair of edges e and f in a graph is independent if e and f do not share a vertex. We show that a graph G is radial planar if G has a radial drawing in which every two independent edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the strong Hanani-Tutte theorem for radial planarity. This characterization yields a very simple algorithm for radial planarity testing.
AU - Fulek, Radoslav
AU - Pelsmajer, Michael
AU - Schaefer, Marcus
ID - 1164
TI - Hanani-Tutte for radial planarity II
VL - 9801
ER -
TY - CONF
AB - We show that c-planarity is solvable in quadratic time for flat clustered graphs with three clusters if the combinatorial embedding of the underlying graph is fixed. In simpler graph-theoretical terms our result can be viewed as follows. Given a graph G with the vertex set partitioned into three parts embedded on a 2-sphere, our algorithm decides if we can augment G by adding edges without creating an edge-crossing so that in the resulting spherical graph the vertices of each part induce a connected sub-graph. We proceed by a reduction to the problem of testing the existence of a perfect matching in planar bipartite graphs. We formulate our result in a slightly more general setting of cyclic clustered graphs, i.e., the simple graph obtained by contracting each cluster, where we disregard loops and multi-edges, is a cycle.
AU - Fulek, Radoslav
ID - 1165
TI - C-planarity of embedded cyclic c-graphs
VL - 9801
ER -
TY - CONF
AB - Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in ℝm. Cohomology and homology groups are well known topological invariants of such sets. Cohomological operations, such as the cup product, provide higher-order algebraic topological invariants, especially important for digital images of dimension higher than 3. If such an operation is determined at the level of simplicial chains [see e.g. González-Díaz, Real, Homology, Homotopy Appl, 2003, 83-93], then it is effectively computable. However, decomposing a cubical complex into a simplicial one deleteriously affects the efficiency of such an approach. In order to avoid this overhead, a direct cubical approach was applied in [Pilarczyk, Real, Adv. Comput. Math., 2015, 253-275] for the cup product in cohomology, and implemented in the ChainCon software package [http://www.pawelpilarczyk.com/chaincon/]. We establish a formula for the Steenrod square operations [see Steenrod, Annals of Mathematics. Second Series, 1947, 290-320] directly at the level of cubical chains, and we prove the correctness of this formula. An implementation of this formula is programmed in C++ within the ChainCon software framework. We provide a few examples and discuss the effectiveness of this approach. One specific application follows from the fact that Steenrod squares yield tests for the topological extension problem: Can a given map A → Sd to a sphere Sd be extended to a given super-complex X of A? In particular, the ROB-SAT problem, which is to decide for a given function f: X → ℝm and a value r > 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the extension problem.
AU - Krcál, Marek
AU - Pilarczyk, Pawel
ID - 1237
TI - Computation of cubical Steenrod squares
VL - 9667
ER -
TY - THES
AB - Motivated by topological Tverberg-type problems in topological combinatorics and by classical
results about embeddings (maps without double points), we study the question whether a finite
simplicial complex K can be mapped into Rd without triple, quadruple, or, more generally, r-fold points (image points with at least r distinct preimages), for a given multiplicity r ≤ 2. In particular, we are interested in maps f : K → Rd that have no global r -fold intersection points, i.e., no r -fold points with preimages in r pairwise disjoint simplices of K , and we seek necessary and sufficient conditions for the existence of such maps.
We present higher-multiplicity analogues of several classical results for embeddings, in particular of the completeness of the Van Kampen obstruction for embeddability of k -dimensional
complexes into R2k , k ≥ 3. Speciffically, we show that under suitable restrictions on the dimensions(viz., if dimK = (r ≥ 1)k and d = rk \ for some k ≥ 3), a well-known deleted product criterion (DPC ) is not only necessary but also sufficient for the existence of maps without global r -fold points. Our main technical tool is a higher-multiplicity version of the classical Whitney trick , by which pairs of isolated r -fold points of opposite sign can be eliminated by local modiffications of the map, assuming codimension d – dimK ≥ 3.
An important guiding idea for our work was that suffciency of the DPC, together with an old
result of Özaydin's on the existence of equivariant maps, might yield an approach to disproving the remaining open cases of the the long-standing topological Tverberg conjecture , i.e., to construct maps from the N -simplex σN to Rd without r-Tverberg points when r not a prime power and
N = (d + 1)(r – 1). Unfortunately, our proof of the sufficiency of the DPC requires codimension d – dimK ≥ 3, which is not satisfied for K = σN .
In 2015, Frick [16] found a very elegant way to overcome this \codimension 3 obstacle" and
to construct the first counterexamples to the topological Tverberg conjecture for all parameters(d; r ) with d ≥ 3r + 1 and r not a prime power, by a reduction1 to a suitable lower-dimensional skeleton, for which the codimension 3 restriction is satisfied and maps without r -Tverberg points exist by Özaydin's result and sufficiency of the DPC.
In this thesis, we present a different construction (which does not use the constraint method) that yields counterexamples for d ≥ 3r , r not a prime power.
AU - Mabillard, Isaac
ID - 1123
TI - Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture
ER -
TY - GEN
AB - We study conditions under which a finite simplicial complex $K$ can be mapped to $\mathbb R^d$ without higher-multiplicity intersections. An almost $r$-embedding is a map $f: K\to \mathbb R^d$ such that the images of any $r$
pairwise disjoint simplices of $K$ do not have a common point. We show that if $r$ is not a prime power and $d\geq 2r+1$, then there is a counterexample to the topological Tverberg conjecture, i.e., there is an almost $r$-embedding of
the $(d+1)(r-1)$-simplex in $\mathbb R^d$. This improves on previous constructions of counterexamples (for $d\geq 3r$) based on a series of papers by M. \"Ozaydin, M. Gromov, P. Blagojevi\'c, F. Frick, G. Ziegler, and the second and fourth present authors. The counterexamples are obtained by proving the following algebraic criterion in codimension 2: If $r\ge3$ and if $K$ is a finite $2(r-1)$-complex then there exists an almost $r$-embedding $K\to \mathbb R^{2r}$ if and only if there exists a general position PL map $f:K\to \mathbb R^{2r}$ such that the algebraic intersection number of the $f$-images of any $r$ pairwise disjoint simplices of $K$ is zero. This result can be restated in terms of cohomological obstructions or equivariant maps, and extends an analogous codimension 3 criterion by the second and fourth authors. As another application we classify ornaments $f:S^3 \sqcup S^3\sqcup S^3\to \mathbb R^5$ up to ornament
concordance. It follows from work of M. Freedman, V. Krushkal and P. Teichner that the analogous criterion for $r=2$ is false. We prove a lemma on singular higher-dimensional Borromean rings, yielding an elementary proof of the counterexample.
AU - Avvakumov, Sergey
AU - Mabillard, Isaac
AU - Skopenkov, A.
AU - Wagner, Uli
ID - 8183
T2 - arXiv
TI - Eliminating higher-multiplicity intersections, III. Codimension 2
ER -
TY - JOUR
AB - We estimate the selection constant in the following geometric selection theorem by Pach: For every positive integer d, there is a constant (Formula presented.) such that whenever (Formula presented.) are n-element subsets of (Formula presented.), we can find a point (Formula presented.) and subsets (Formula presented.) for every i∈[d+1], each of size at least cdn, such that p belongs to all rainbowd-simplices determined by (Formula presented.) simplices with one vertex in each Yi. We show a super-exponentially decreasing upper bound (Formula presented.). The ideas used in the proof of the upper bound also help us to prove Pach’s theorem with (Formula presented.), which is a lower bound doubly exponentially decreasing in d (up to some polynomial in the exponent). For comparison, Pach’s original approach yields a triply exponentially decreasing lower bound. On the other hand, Fox, Pach, and Suk recently obtained a hypergraph density result implying a proof of Pach’s theorem with (Formula presented.). In our construction for the upper bound, we use the fact that the minimum solid angle of every d-simplex is super-exponentially small. This fact was previously unknown and might be of independent interest. For the lower bound, we improve the ‘separation’ part of the argument by showing that in one of the key steps only d+1 separations are necessary, compared to 2d separations in the original proof. We also provide a measure version of Pach’s theorem.
AU - Karasev, Roman
AU - Kynčl, Jan
AU - Paták, Pavel
AU - Patakova, Zuzana
AU - Tancer, Martin
ID - 1688
IS - 3
JF - Discrete & Computational Geometry
TI - Bounds for Pach's selection theorem and for the minimum solid angle in a simplex
VL - 54
ER -
TY - JOUR
AB - How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart). Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and vice-versa. This implies a conjecture by Przytycka and Przytycki (Graph structure theory. Contemporary Mathematics, vol. 147, 1993), a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov’s systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions. Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length O(g^(3/2)n^(1/2)) for any triangulated combinatorial surface of genus g with n triangles, and describe an O(gn)-time algorithm to compute such a decomposition. Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.
AU - Colin De Verdière, Éric
AU - Hubard, Alfredo
AU - De Mesmay, Arnaud N
ID - 1730
IS - 3
JF - Discrete & Computational Geometry
TI - Discrete systolic inequalities and decompositions of triangulated surfaces
VL - 53
ER -
TY - CONF
AB - The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r > 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f, f' from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.
AU - Franek, Peter
AU - Krcál, Marek
ID - 1510
TI - On computability and triviality of well groups
VL - 34
ER -
TY - CONF
AB - The fact that the complete graph K_5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M if and only if (n-3)(n-4) is at most 6b_1(M), where b_1(M) is the first Z_2-Betti number of M. On the other hand, Van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_{n+1}) embeds in R^{2k} if and only if n is less or equal to 2k+2. Two decades ago, Kuhnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k-1)-connected 2k-manifold with kth Z_2-Betti number b_k only if the following generalized Heawood inequality holds: binom{n-k-1}{k+1} is at most binom{2k+1}{k+1} b_k. This is a common generalization of the case of graphs on surfaces as well as the Van Kampen--Flores theorem. In the spirit of Kuhnel's conjecture, we prove that if the k-skeleton of the n-simplex embeds in a 2k-manifold with kth Z_2-Betti number b_k, then n is at most 2b_k binom{2k+2}{k} + 2k + 5. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k-1)-connected. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.
AU - Goaoc, Xavier
AU - Mabillard, Isaac
AU - Paták, Pavel
AU - Patakova, Zuzana
AU - Tancer, Martin
AU - Wagner, Uli
ID - 1511
TI - On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability result
VL - 34
ER -
TY - CONF
AB - We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b,d) such that the following holds. If F is a finite family of subsets of R^d such that the ith reduced Betti number (with Z_2 coefficients in singular homology) of the intersection of any proper subfamily G of F is at most b for every non-negative integer i less or equal to (d-1)/2, then F has Helly number at most h(b,d). These topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map from C_*(K) to C_*(R^d). Both techniques are of independent interest.
AU - Goaoc, Xavier
AU - Paták, Pavel
AU - Patakova, Zuzana
AU - Tancer, Martin
AU - Wagner, Uli
ID - 1512
TI - Bounding Helly numbers via Betti numbers
VL - 34
ER -
TY - CONF
AB - A drawing of a graph G is radial if the vertices of G are placed on concentric circles C1, . . . , Ck with common center c, and edges are drawn radially: every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing- free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. We show that a graph G is radial planar if G has a radial drawing in which every two edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Tóth.
AU - Fulek, Radoslav
AU - Pelsmajer, Michael
AU - Schaefer, Marcus
ID - 1595
TI - Hanani-Tutte for radial planarity
VL - 9411
ER -
TY - CONF
AB - Let C={C1,...,Cn} denote a collection of translates of a regular convex k-gon in the plane with the stacking order. The collection C forms a visibility clique if for everyi < j the intersection Ci and (Ci ∩ Cj)\⋃i<l<jCl =∅.elements that are stacked between them, i.e., We show that if C forms a visibility clique its size is bounded from above by O(k4) thereby improving the upper bound of 22k from the aforementioned paper. We also obtain an upper bound of 22(k/2)+2 on the size of a visibility clique for homothetes of a convex (not necessarily regular) k-gon.
AU - Fulek, Radoslav
AU - Radoičić, Radoš
ID - 1596
TI - Vertical visibility among parallel polygons in three dimensions
VL - 9411
ER -
TY - JOUR
AB - The Hanani-Tutte theorem is a classical result proved for the first time in the 1930s that characterizes planar graphs as graphs that admit a drawing in the plane in which every pair of edges not sharing a vertex cross an even number of times. We generalize this result to clustered graphs with two disjoint clusters, and show that a straightforward extension to flat clustered graphs with three or more disjoint clusters is not possible. For general clustered graphs we show a variant of the Hanani-Tutte theorem in the case when each cluster induces a connected subgraph. Di Battista and Frati proved that clustered planarity of embedded clustered graphs whose every face is incident to at most five vertices can be tested in polynomial time. We give a new and short proof of this result, using the matroid intersection algorithm.
AU - Fulek, Radoslav
AU - Kynčl, Jan
AU - Malinovič, Igor
AU - Pálvölgyi, Dömötör
ID - 1642
IS - 4
JF - Electronic Journal of Combinatorics
TI - Clustered planarity testing revisited
VL - 22
ER -
TY - JOUR
AB - We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f:K→ ℝn on a finite simplicial complex K and α > 0, it holds that each function g: K → ℝn such that ||g - f || ∞ < α, has a root in K. Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed n, assuming dimK ≤ 2n - 3. This is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis. Via a reverse reduction, we prove that the problem is undecidable when dim K > 2n - 2, where the threshold comes from the stable range in homotopy theory. For the lucidity of our exposition, we focus on the setting when f is simplexwise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.
AU - Franek, Peter
AU - Krcál, Marek
ID - 1682
IS - 4
JF - Journal of the ACM
TI - Robust satisfiability of systems of equations
VL - 62
ER -
TY - CONF
AB - Given a graph G cellularly embedded on a surface Σ of genus g, a cut graph is a subgraph of G such that cutting Σ along G yields a topological disk. We provide a fixed parameter tractable approximation scheme for the problem of computing the shortest cut graph, that is, for any ε > 0, we show how to compute a (1 + ε) approximation of the shortest cut graph in time f(ε, g)n3.
Our techniques first rely on the computation of a spanner for the problem using the technique of brick decompositions, to reduce the problem to the case of bounded tree-width. Then, to solve the bounded tree-width case, we introduce a variant of the surface-cut decomposition of Rué, Sau and Thilikos, which may be of independent interest.
AU - Cohen Addad, Vincent
AU - De Mesmay, Arnaud N
ID - 1685
TI - A fixed parameter tractable approximation scheme for the optimal cut graph of a surface
VL - 9294
ER -
TY - JOUR
AB - We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on 2n vertices are bounded by O(n3) and O(n10), in the convex and general case, respectively. We then apply similar methods to prove an (Formula presented.) upper bound on the Ramsey number of a path with n ordered vertices.
AU - Cibulka, Josef
AU - Gao, Pu
AU - Krcál, Marek
AU - Valla, Tomáš
AU - Valtr, Pavel
ID - 1842
IS - 1
JF - Discrete & Computational Geometry
TI - On the geometric ramsey number of outerplanar graphs
VL - 53
ER -
TY - JOUR
AB - A result of Boros and Füredi (d = 2) and of Bárány (arbitrary d) asserts that for every d there exists cd > 0 such that for every n-point set P ⊂ ℝd, some point of ℝd is covered by at least (Formula presented.) of the d-simplices spanned by the points of P. The largest possible value of cd has been the subject of ongoing research. Recently Gromov improved the existing lower bounds considerably by introducing a new, topological proof method. We provide an exposition of the combinatorial component of Gromov's approach, in terms accessible to combinatorialists and discrete geometers, and we investigate the limits of his method. In particular, we give tighter bounds on the cofilling profiles for the (n - 1)-simplex. These bounds yield a minor improvement over Gromov's lower bounds on cd for large d, but they also show that the room for further improvement through the cofilling profiles alone is quite small. We also prove a slightly better lower bound for c3 by an approach using an additional structure besides the cofilling profiles. We formulate a combinatorial extremal problem whose solution might perhaps lead to a tight lower bound for cd.
AU - Matoušek, Jiří
AU - Wagner, Uli
ID - 2154
IS - 1
JF - Discrete & Computational Geometry
TI - On Gromov's method of selecting heavily covered points
VL - 52
ER -
TY - CONF
AB - We show that the following algorithmic problem is decidable: given a 2-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in ℝ3? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold X into the 3-sphere S3. The main step, which allows us to simplify X and recurse, is in proving that if X can be embedded in S3, then there is also an embedding in which X has a short meridian, i.e., an essential curve in the boundary of X bounding a disk in S3 nX with length bounded by a computable function of the number of tetrahedra of X.
AU - Matoušek, Jiří
AU - Sedgwick, Eric
AU - Tancer, Martin
AU - Wagner, Uli
ID - 2157
T2 - Proceedings of the Annual Symposium on Computational Geometry
TI - Embeddability in the 3 sphere is decidable
ER -
TY - JOUR
AB - Given topological spaces X,Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X→ Y. We consider a computational version, where X,Y are given as finite simplicial complexes, and the goal is to compute [X,Y], that is, all homotopy classes of suchmaps.We solve this problem in the stable range, where for some d ≥ 2, we have dim X ≤ 2d-2 and Y is (d-1)-connected; in particular, Y can be the d-dimensional sphere Sd. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X = S1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y. In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A ⊂ X and a map A→ Y and ask whether it extends to a map X → Y, or computing the Z2-index-everything in the stable range. Outside the stable range, the extension problem is undecidable.
AU - Čadek, Martin
AU - Krcál, Marek
AU - Matoušek, Jiří
AU - Sergeraert, Francis
AU - Vokřínek, Lukáš
AU - Wagner, Uli
ID - 2184
IS - 3
JF - Journal of the ACM
TI - Computing all maps into a sphere
VL - 61
ER -
TY - GEN
AU - Huszár, Kristóf
AU - Rolinek, Michal
ID - 7038
TI - Playful Math - An introduction to mathematical games
ER -
TY - CONF
AB - Motivated by topological Tverberg-type problems, we consider multiple (double, triple, and higher multiplicity) selfintersection points of maps from finite simplicial complexes (compact polyhedra) into ℝd and study conditions under which such multiple points can be eliminated. The most classical case is that of embeddings (i.e., maps without double points) of a κ-dimensional complex K into ℝ2κ. For this problem, the work of van Kampen, Shapiro, and Wu provides an efficiently testable necessary condition for embeddability (namely, vanishing of the van Kampen ob-struction). For κ ≥ 3, the condition is also sufficient, and yields a polynomial-time algorithm for deciding embeddability: One starts with an arbitrary map f : K→ℝ2κ, which generically has finitely many double points; if k ≥ 3 and if the obstruction vanishes then one can successively remove these double points by local modifications of the map f. One of the main tools is the famous Whitney trick that permits eliminating pairs of double points of opposite intersection sign. We are interested in generalizing this approach to intersection points of higher multiplicity. We call a point y 2 ℝd an r-fold Tverberg point of a map f : Kκ →ℝd if y lies in the intersection f(σ1)∩. ∩f(σr) of the images of r pairwise disjoint simplices of K. The analogue of (non-)embeddability that we study is the problem Tverbergκ r→d: Given a κ-dimensional complex K, does it satisfy a Tverberg-type theorem with parameters r and d, i.e., does every map f : K κ → ℝd have an r-fold Tverberg point? Here, we show that for fixed r, κ and d of the form d = rm and k = (r-1)m, m ≥ 3, there is a polynomial-time algorithm for deciding this (based on the vanishing of a cohomological obstruction, as in the case of embeddings). Our main tool is an r-fold analogue of the Whitney trick: Given r pairwise disjoint simplices of K such that the intersection of their images contains two r-fold Tverberg points y+ and y- of opposite intersection sign, we can eliminate y+ and y- by a local isotopy of f. In a subsequent paper, we plan to develop this further and present a generalization of the classical Haeiger-Weber Theorem (which yields a necessary and sufficient condition for embeddability of κ-complexes into ℝd for a wider range of dimensions) to intersection points of higher multiplicity.
AU - Mabillard, Isaac
AU - Wagner, Uli
ID - 2159
T2 - Proceedings of the Annual Symposium on Computational Geometry
TI - Eliminating Tverberg points, I. An analogue of the Whitney trick
ER -
TY - CONF
AB - We consider several basic problems of algebraic topology, with connections to combinatorial and geometric questions, from the point of view of computational complexity. The extension problem asks, given topological spaces X; Y , a subspace A ⊆ X, and a (continuous) map f : A → Y , whether f can be extended to a map X → Y . For computational purposes, we assume that X and Y are represented as finite simplicial complexes, A is a subcomplex of X, and f is given as a simplicial map. In this generality the problem is undecidable, as follows from Novikov's result from the 1950s on uncomputability of the fundamental group π1(Y ). We thus study the problem under the assumption that, for some k ≥ 2, Y is (k - 1)-connected; informally, this means that Y has \no holes up to dimension k-1" (a basic example of such a Y is the sphere Sk). We prove that, on the one hand, this problem is still undecidable for dimX = 2k. On the other hand, for every fixed k ≥ 2, we obtain an algorithm that solves the extension problem in polynomial time assuming Y (k - 1)-connected and dimX ≤ 2k - 1. For dimX ≤ 2k - 2, the algorithm also provides a classification of all extensions up to homotopy (continuous deformation). This relies on results of our SODA 2012 paper, and the main new ingredient is a machinery of objects with polynomial-time homology, which is a polynomial-time analog of objects with effective homology developed earlier by Sergeraert et al. We also consider the computation of the higher homotopy groups πk(Y ), k ≥ 2, for a 1-connected Y . Their computability was established by Brown in 1957; we show that πk(Y ) can be computed in polynomial time for every fixed k ≥ 2. On the other hand, Anick proved in 1989 that computing πk(Y ) is #P-hard if k is a part of input, where Y is a cell complex with certain rather compact encoding. We strengthen his result to #P-hardness for Y given as a simplicial complex.
AU - Čadek, Martin
AU - Krcál, Marek
AU - Matoušek, Jiří
AU - Vokřínek, Lukáš
AU - Wagner, Uli
ID - 2807
T2 - 45th Annual ACM Symposium on theory of computing
TI - Extending continuous maps: Polynomiality and undecidability
ER -
TY - CONF
AB - We consider two systems (α1,...,αm) and (β1,...,βn) of curves drawn on a compact two-dimensional surface ℳ with boundary. Each αi and each βj is either an arc meeting the boundary of ℳ at its two endpoints, or a closed curve. The αi are pairwise disjoint except for possibly sharing endpoints, and similarly for the βj. We want to "untangle" the βj from the αi by a self-homeomorphism of ℳ; more precisely, we seek an homeomorphism φ: ℳ → ℳ fixing the boundary of ℳ pointwise such that the total number of crossings of the αi with the φ(βj) is as small as possible. This problem is motivated by an application in the algorithmic theory of embeddings and 3-manifolds. We prove that if ℳ is planar, i.e., a sphere with h ≥ 0 boundary components ("holes"), then O(mn) crossings can be achieved (independently of h), which is asymptotically tight, as an easy lower bound shows. In general, for an arbitrary (orientable or nonorientable) surface ℳ with h holes and of (orientable or nonorientable) genus g ≥ 0, we obtain an O((m + n)4) upper bound, again independent of h and g.
AU - Matoušek, Jiří
AU - Sedgwick, Eric
AU - Tancer, Martin
AU - Wagner, Uli
ID - 2244
TI - Untangling two systems of noncrossing curves
VL - 8242
ER -