@inproceedings{14888, abstract = {A face in a curve arrangement is called popular if it is bounded by the same curve multiple times. Motivated by the automatic generation of curved nonogram puzzles, we investigate possibilities to eliminate the popular faces in an arrangement by inserting a single additional curve. This turns out to be NP-hard; however, it becomes tractable when the number of popular faces is small: We present a probabilistic FPT-approach in the number of popular faces.}, author = {De Nooijer, Phoebe and Terziadis, Soeren and Weinberger, Alexandra and Masárová, Zuzana and Mchedlidze, Tamara and Löffler, Maarten and Rote, Günter}, booktitle = {31st International Symposium on Graph Drawing and Network Visualization}, isbn = {9783031492747}, issn = {1611-3349}, location = {Isola delle Femmine, Palermo, Italy}, pages = {18--33}, publisher = {Springer Nature}, title = {{Removing popular faces in curve arrangements}}, doi = {10.1007/978-3-031-49275-4_2}, volume = {14466}, year = {2024}, } @inproceedings{15012, abstract = {We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on n vertices cannot be decomposed into fewer than n-1 star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs.}, author = {Pach, János and Saghafian, Morteza and Schnider, Patrick}, booktitle = {31st International Symposium on Graph Drawing and Network Visualization}, isbn = {9783031492716}, issn = {16113349}, location = {Isola delle Femmine, Palermo, Italy}, pages = {339--346}, publisher = {Springer Nature}, title = {{Decomposition of geometric graphs into star-forests}}, doi = {10.1007/978-3-031-49272-3_23}, volume = {14465}, year = {2024}, } @inproceedings{15093, abstract = {We present a dynamic data structure for maintaining the persistent homology of a time series of real numbers. The data structure supports local operations, including the insertion and deletion of an item and the cutting and concatenating of lists, each in time O(log n + k), in which n counts the critical items and k the changes in the augmented persistence diagram. To achieve this, we design a tailor-made tree structure with an unconventional representation, referred to as banana tree, which may be useful in its own right.}, author = {Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Henzinger, Monika H and Ost, Lara}, booktitle = {Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)}, editor = {Woodruff, David P.}, location = {Alexandria, VA, USA}, pages = {243 -- 295}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Dynamically maintaining the persistent homology of time series}}, doi = {10.1137/1.9781611977912.11}, year = {2024}, } @unpublished{15091, abstract = {Motivated by applications in the medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided.}, author = {Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {arXiv}, title = {{Chromatic alpha complexes}}, year = {2024}, } @inproceedings{13048, abstract = {In this paper we introduce a pruning of the medial axis called the (λ,α)-medial axis (axλα). We prove that the (λ,α)-medial axis of a set K is stable in a Gromov-Hausdorff sense under weak assumptions. More formally we prove that if K and K′ are close in the Hausdorff (dH) sense then the (λ,α)-medial axes of K and K′ are close as metric spaces, that is the Gromov-Hausdorff distance (dGH) between the two is 1/4-Hölder in the sense that dGH (axλα(K),axλα(K′)) ≲ dH(K,K′)1/4. The Hausdorff distance between the two medial axes is also bounded, by dH (axλα(K),λα(K′)) ≲ dH(K,K′)1/2. These quantified stability results provide guarantees for practical computations of medial axes from approximations. Moreover, they provide key ingredients for studying the computability of the medial axis in the context of computable analysis.}, author = {Lieutier, André and Wintraecken, Mathijs}, booktitle = {Proceedings of the 55th Annual ACM Symposium on Theory of Computing}, isbn = {9781450399135}, location = {Orlando, FL, United States}, pages = {1768--1776}, publisher = {Association for Computing Machinery}, title = {{Hausdorff and Gromov-Hausdorff stable subsets of the medial axis}}, doi = {10.1145/3564246.3585113}, year = {2023}, } @article{12086, abstract = {We present a simple algorithm for computing higher-order Delaunay mosaics that works in Euclidean spaces of any finite dimensions. The algorithm selects the vertices of the order-k mosaic from incrementally constructed lower-order mosaics and uses an algorithm for weighted first-order Delaunay mosaics as a black-box to construct the order-k mosaic from its vertices. Beyond this black-box, the algorithm uses only combinatorial operations, thus facilitating easy implementation. We extend this algorithm to compute higher-order α-shapes and provide open-source implementations. We present experimental results for properties of higher-order Delaunay mosaics of random point sets.}, author = {Edelsbrunner, Herbert and Osang, Georg F}, issn = {1432-0541}, journal = {Algorithmica}, pages = {277--295}, publisher = {Springer Nature}, title = {{A simple algorithm for higher-order Delaunay mosaics and alpha shapes}}, doi = {10.1007/s00453-022-01027-6}, volume = {85}, year = {2023}, } @article{12287, abstract = {We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of A is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use.}, author = {Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit and Wintraecken, Mathijs}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, keywords = {Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Theoretical Computer Science}, pages = {156--191}, publisher = {Springer Nature}, title = {{Local criteria for triangulating general manifolds}}, doi = {10.1007/s00454-022-00431-7}, volume = {69}, year = {2023}, } @inproceedings{12548, abstract = {The limited exchange between human communities is a key factor in preventing the spread of COVID-19. This paper introduces a digital framework that combines an integration of real mobility data at the country scale with a series of modeling techniques and visual capabilities that highlight mobility patterns before and during the pandemic. The findings not only significantly exhibit mobility trends and different degrees of similarities at regional and local levels but also provide potential insight into the emergence of a pandemic on human behavior patterns and their likely socio-economic impacts.}, author = {Forghani, Mohammad and Claramunt, Christophe and Karimipour, Farid and Heiler, Georg}, booktitle = {2022 IEEE International Conference on Data Mining Workshops}, issn = {2375-9259}, location = {Orlando, FL, United States}, publisher = {Institute of Electrical and Electronics Engineers}, title = {{Visual analytics of mobility network changes observed using mobile phone data during COVID-19 pandemic}}, doi = {10.1109/icdmw58026.2022.00093}, year = {2023}, } @article{12544, abstract = {Geometry is crucial in our efforts to comprehend the structures and dynamics of biomolecules. For example, volume, surface area, and integrated mean and Gaussian curvature of the union of balls representing a molecule are used to quantify its interactions with the water surrounding it in the morphometric implicit solvent models. The Alpha Shape theory provides an accurate and reliable method for computing these geometric measures. In this paper, we derive homogeneous formulas for the expressions of these measures and their derivatives with respect to the atomic coordinates, and we provide algorithms that implement them into a new software package, AlphaMol. The only variables in these formulas are the interatomic distances, making them insensitive to translations and rotations. AlphaMol includes a sequential algorithm and a parallel algorithm. In the parallel version, we partition the atoms of the molecule of interest into 3D rectangular blocks, using a kd-tree algorithm. We then apply the sequential algorithm of AlphaMol to each block, augmented by a buffer zone to account for atoms whose ball representations may partially cover the block. The current parallel version of AlphaMol leads to a 20-fold speed-up compared to an independent serial implementation when using 32 processors. For instance, it takes 31 s to compute the geometric measures and derivatives of each atom in a viral capsid with more than 26 million atoms on 32 Intel processors running at 2.7 GHz. The presence of the buffer zones, however, leads to redundant computations, which ultimately limit the impact of using multiple processors. AlphaMol is available as an OpenSource software.}, author = {Koehl, Patrice and Akopyan, Arseniy and Edelsbrunner, Herbert}, issn = {1549-960X}, journal = {Journal of Chemical Information and Modeling}, number = {3}, pages = {973--985}, publisher = {American Chemical Society}, title = {{Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives}}, doi = {10.1021/acs.jcim.2c01346}, volume = {63}, year = {2023}, } @article{12764, abstract = {We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi cell corresponding to the singularity. We divide polyhedral surfaces into discrete conformal classes using a generalization of discrete conformal equivalence pioneered by Feng Luo. We subsequently show that, in every discrete conformal class, there exists a polyhedral surface with constant discrete Gaussian curvature. We also provide explicit examples to demonstrate that this surface is in general not unique.}, author = {Kourimska, Hana}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {123--153}, publisher = {Springer Nature}, title = {{Discrete yamabe problem for polyhedral surfaces}}, doi = {10.1007/s00454-023-00484-2}, volume = {70}, year = {2023}, } @article{12709, abstract = {Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.}, author = {Corbet, René and Kerber, Michael and Lesnick, Michael and Osang, Georg F}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {376--405}, publisher = {Springer Nature}, title = {{Computing the multicover bifiltration}}, doi = {10.1007/s00454-022-00476-8}, volume = {70}, year = {2023}, } @article{12763, abstract = {Kleinjohann (Archiv der Mathematik 35(1):574–582, 1980; Mathematische Zeitschrift 176(3), 327–344, 1981) and Bangert (Archiv der Mathematik 38(1):54–57, 1982) extended the reach rch(S) from subsets S of Euclidean space to the reach rchM(S) of subsets S of Riemannian manifolds M, where M is smooth (we’ll assume at least C3). Bangert showed that sets of positive reach in Euclidean space and Riemannian manifolds are very similar. In this paper we introduce a slight variant of Kleinjohann’s and Bangert’s extension and quantify the similarity between sets of positive reach in Euclidean space and Riemannian manifolds in a new way: Given p∈M and q∈S, we bound the local feature size (a local version of the reach) of its lifting to the tangent space via the inverse exponential map (exp−1p(S)) at q, assuming that rchM(S) and the geodesic distance dM(p,q) are bounded. These bounds are motivated by the importance of the reach and local feature size to manifold learning, topological inference, and triangulating manifolds and the fact that intrinsic approaches circumvent the curse of dimensionality.}, author = {Boissonnat, Jean Daniel and Wintraecken, Mathijs}, issn = {2367-1734}, journal = {Journal of Applied and Computational Topology}, pages = {619--641}, publisher = {Springer Nature}, title = {{The reach of subsets of manifolds}}, doi = {10.1007/s41468-023-00116-x}, volume = {7}, year = {2023}, } @article{12960, abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e., submanifolds of Rd defined as the zero set of some multivariate multivalued smooth function f:Rd→Rd−n, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M=f−1(0) is to consider its piecewise linear (PL) approximation M^ based on a triangulation T of the ambient space Rd. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ=1/D (and unavoidably exponential in n). Since it is known that for δ=Ω(d2.5), M^ is O(D2)-close and isotopic to M , our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M^ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art. }, author = {Boissonnat, Jean Daniel and Kachanovich, Siargey and Wintraecken, Mathijs}, issn = {1095-7111}, journal = {SIAM Journal on Computing}, number = {2}, pages = {452--486}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Tracing isomanifolds in Rd in time polynomial in d using Coxeter–Freudenthal–Kuhn triangulations}}, doi = {10.1137/21M1412918}, volume = {52}, year = {2023}, } @article{13134, abstract = {We propose a characterization of discrete analytical spheres, planes and lines in the body-centered cubic (BCC) grid, both in the Cartesian and in the recently proposed alternative compact coordinate system, in which each integer triplet addresses some voxel in the grid. We define spheres and planes through double Diophantine inequalities and investigate their relevant topological features, such as functionality or the interrelation between the thickness of the objects and their connectivity and separation properties. We define lines as the intersection of planes. The number of the planes (up to six) is equal to the number of the pairs of faces of a BCC voxel that are parallel to the line.}, author = {Čomić, Lidija and Largeteau-Skapin, Gaëlle and Zrour, Rita and Biswas, Ranita and Andres, Eric}, issn = {0031-3203}, journal = {Pattern Recognition}, number = {10}, publisher = {Elsevier}, title = {{Discrete analytical objects in the body-centered cubic grid}}, doi = {10.1016/j.patcog.2023.109693}, volume = {142}, year = {2023}, } @article{14557, abstract = {Motivated by a problem posed in [10], we investigate the closure operators of the category SLatt of join semilattices and its subcategory SLattO of join semilattices with bottom element. In particular, we show that there are only finitely many closure operators of both categories, and provide a complete classification. We use this result to deduce the known fact that epimorphisms of SLatt and SLattO are surjective. We complement the paper with two different proofs of this result using either generators or Isbell’s zigzag theorem.}, author = {Dikranjan, D. and Giordano Bruno, A. and Zava, Nicolò}, issn = {1727-933X}, journal = {Quaestiones Mathematicae}, number = {S1}, pages = {191--221}, publisher = {Taylor & Francis}, title = {{Epimorphisms and closure operators of categories of semilattices}}, doi = {10.2989/16073606.2023.2247731}, volume = {46}, year = {2023}, } @article{14345, abstract = {For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970)).}, author = {Edelsbrunner, Herbert and Garber, Alexey and Ghafari, Mohadese and Heiss, Teresa and Saghafian, Morteza}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, publisher = {Springer Nature}, title = {{On angles in higher order Brillouin tessellations and related tilings in the plane}}, doi = {10.1007/s00454-023-00566-1}, year = {2023}, } @article{14464, abstract = {Given a triangle Δ, we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of Δ with respect to area and perimeter. This problem was initially posed by Nandakumar [17, 22] and was first studied by Kiss, Pach, and Somlai [13], who showed that if Δ′ is the smallest area isosceles triangle containing Δ, then Δ′ and Δ share a side and an angle. In the present paper, we prove that for any triangle Δ, every maximum area isosceles triangle embedded in Δ and every maximum perimeter isosceles triangle embedded in Δ shares a side and an angle with Δ. Somewhat surprisingly, the case of minimum perimeter enclosing triangles is different: there are infinite families of triangles Δ whose minimum perimeter isosceles containers do not share a side and an angle with Δ.}, author = {Ambrus, Áron and Csikós, Mónika and Kiss, Gergely and Pach, János and Somlai, Gábor}, issn = {1793-6373}, journal = {International Journal of Foundations of Computer Science}, number = {7}, pages = {737--760}, publisher = {World Scientific Publishing}, title = {{Optimal embedded and enclosing isosceles triangles}}, doi = {10.1142/S012905412342008X}, volume = {34}, year = {2023}, } @article{12833, abstract = {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.}, author = {Biniaz, Ahmad and Jain, Kshitij and Lubiw, Anna and Masárová, Zuzana and Miltzow, Tillmann and Mondal, Debajyoti and Naredla, Anurag Murty and Tkadlec, Josef and Turcotte, Alexi}, issn = {1365-8050}, journal = {Discrete Mathematics and Theoretical Computer Science}, number = {2}, publisher = {EPI Sciences}, title = {{Token swapping on trees}}, doi = {10.46298/DMTCS.8383}, volume = {24}, year = {2023}, } @article{14739, abstract = {Attempts to incorporate topological information in supervised learning tasks have resulted in the creation of several techniques for vectorizing persistent homology barcodes. In this paper, we study thirteen such methods. Besides describing an organizational framework for these methods, we comprehensively benchmark them against three well-known classification tasks. Surprisingly, we discover that the best-performing method is a simple vectorization, which consists only of a few elementary summary statistics. Finally, we provide a convenient web application which has been designed to facilitate exploration and experimentation with various vectorization methods.}, author = {Ali, Dashti and Asaad, Aras and Jimenez, Maria-Jose and Nanda, Vidit and Paluzo-Hidalgo, Eduardo and Soriano Trigueros, Manuel}, issn = {1939-3539}, journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence}, keywords = {Applied Mathematics, Artificial Intelligence, Computational Theory and Mathematics, Computer Vision and Pattern Recognition, Software}, number = {12}, pages = {14069--14080}, publisher = {IEEE}, title = {{A survey of vectorization methods in topological data analysis}}, doi = {10.1109/tpami.2023.3308391}, volume = {45}, year = {2023}, } @article{13165, abstract = {A graph G=(V, E) is called fully regular if for every independent set I c V, the number of vertices in V\I that are not connected to any element of I depends only on the size of I. A linear ordering of the vertices of G is called successive if for every i, the first i vertices induce a connected subgraph of G. We give an explicit formula for the number of successive vertex orderings of a fully regular graph. As an application of our results, we give alternative proofs of two theorems of Stanley and Gao & Peng, determining the number of linear edge orderings of complete graphs and complete bipartite graphs, respectively, with the property that the first i edges induce a connected subgraph. As another application, we give a simple product formula for the number of linear orderings of the hyperedges of a complete 3-partite 3-uniform hypergraph such that, for every i, the first i hyperedges induce a connected subgraph. We found similar formulas for complete (non-partite) 3-uniform hypergraphs and in another closely related case, but we managed to verify them only when the number of vertices is small.}, author = {Fang, Lixing and Huang, Hao and Pach, János and Tardos, Gábor and Zuo, Junchi}, issn = {1096-0899}, journal = {Journal of Combinatorial Theory. Series A}, number = {10}, publisher = {Elsevier}, title = {{Successive vertex orderings of fully regular graphs}}, doi = {10.1016/j.jcta.2023.105776}, volume = {199}, year = {2023}, }