TY - JOUR AB - We consider a natural problem dealing with weighted packet selection across a rechargeable link, which e.g., finds applications in cryptocurrency networks. The capacity of a link (u, v) is determined by how many nodes u and v allocate for this link. Specifically, the input is a finite ordered sequence of packets that arrive in both directions along a link. Given (u, v) and a packet of weight x going from u to v, node u can either accept or reject the packet. If u accepts the packet, the capacity on link (u, v) decreases by x. Correspondingly, v's capacity on increases by x. If a node rejects the packet, this will entail a cost affinely linear in the weight of the packet. A link is “rechargeable” in the sense that the total capacity of the link has to remain constant, but the allocation of capacity at the ends of the link can depend arbitrarily on the nodes' decisions. The goal is to minimise the sum of the capacity injected into the link and the cost of rejecting packets. We show that the problem is NP-hard, but can be approximated efficiently with a ratio of (1+E) . (1+3) for some arbitrary E>0. AU - Schmid, Stefan AU - Svoboda, Jakub AU - Yeo, Michelle X ID - 14820 JF - Theoretical Computer Science KW - General Computer Science KW - Theoretical Computer Science SN - 0304-3975 TI - Weighted packet selection for rechargeable links in cryptocurrency networks: Complexity and approximation VL - 989 ER - TY - JOUR AB - 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. AU - Boissonnat, Jean-Daniel AU - Dyer, Ramsay AU - Ghosh, Arijit AU - Wintraecken, Mathijs ID - 12287 JF - Discrete & Computational Geometry KW - Computational Theory and Mathematics KW - Discrete Mathematics and Combinatorics KW - Geometry and Topology KW - Theoretical Computer Science SN - 0179-5376 TI - Local criteria for triangulating general manifolds VL - 69 ER - TY - JOUR AB - A shared-memory counter is a widely-used and well-studied concurrent object. It supports two operations: An Inc operation that increases its value by 1 and a Read operation that returns its current value. In Jayanti et al (SIAM J Comput, 30(2), 2000), Jayanti, Tan and Toueg proved a linear lower bound on the worst-case step complexity of obstruction-free implementations, from read-write registers, of a large class of shared objects that includes counters. The lower bound leaves open the question of finding counter implementations with sub-linear amortized step complexity. In this work, we address this gap. We show that n-process, wait-free and linearizable counters can be implemented from read-write registers with O(log2n) amortized step complexity. This is the first counter algorithm from read-write registers that provides sub-linear amortized step complexity in executions of arbitrary length. Since a logarithmic lower bound on the amortized step complexity of obstruction-free counter implementations exists, our upper bound is within a logarithmic factor of the optimal. The worst-case step complexity of the construction remains linear, which is optimal. This is obtained thanks to a new max register construction with O(logn) amortized step complexity in executions of arbitrary length in which the value stored in the register does not grow too quickly. We then leverage an existing counter algorithm by Aspnes, Attiya and Censor-Hillel [1] in which we “plug” our max register implementation to show that it remains linearizable while achieving O(log2n) amortized step complexity. AU - Baig, Mirza Ahad AU - Hendler, Danny AU - Milani, Alessia AU - Travers, Corentin ID - 12164 JF - Distributed Computing KW - Computational Theory and Mathematics KW - Computer Networks and Communications KW - Hardware and Architecture KW - Theoretical Computer Science SN - 0178-2770 TI - Long-lived counters with polylogarithmic amortized step complexity VL - 36 ER - TY - JOUR AB - We consider the almost-sure (a.s.) termination problem for probabilistic programs, which are a stochastic extension of classical imperative programs. Lexicographic ranking functions provide a sound and practical approach for termination of non-probabilistic programs, and their extension to probabilistic programs is achieved via lexicographic ranking supermartingales (LexRSMs). However, LexRSMs introduced in the previous work have a limitation that impedes their automation: all of their components have to be non-negative in all reachable states. This might result in a LexRSM not existing even for simple terminating programs. Our contributions are twofold. First, we introduce a generalization of LexRSMs that allows for some components to be negative. This standard feature of non-probabilistic termination proofs was hitherto not known to be sound in the probabilistic setting, as the soundness proof requires a careful analysis of the underlying stochastic process. Second, we present polynomial-time algorithms using our generalized LexRSMs for proving a.s. termination in broad classes of linear-arithmetic programs. AU - Chatterjee, Krishnendu AU - Kafshdar Goharshady, Ehsan AU - Novotný, Petr AU - Zárevúcky, Jiří AU - Zikelic, Dorde ID - 14778 IS - 2 JF - Formal Aspects of Computing KW - Theoretical Computer Science KW - Software SN - 0934-5043 TI - On lexicographic proof rules for probabilistic termination VL - 35 ER - TY - JOUR AB - We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding Gelfand–Naimark–Segal Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite-volume Hamiltonians need not have a spectral gap. AU - Henheik, Sven Joscha AU - Teufel, Stefan ID - 10643 JF - Forum of Mathematics, Sigma KW - computational mathematics KW - discrete mathematics and combinatorics KW - geometry and topology KW - mathematical physics KW - statistics and probability KW - algebra and number theory KW - theoretical computer science KW - analysis TI - Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk VL - 10 ER - TY - JOUR AB - Given a finite point set P in general position in the plane, a full triangulation of P is a maximal straight-line embedded plane graph on P. A partial triangulation of 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 (called edge flip), 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 edge flip graph is defined with full triangulations as vertices, and edge flips determining the adjacencies. Lawson showed in the early seventies that these graphs are connected. The goal of this paper is to investigate the structure of these graphs, with emphasis on their vertex connectivity. For sets P of n points in the plane in general position, we show that the edge flip graph is ⌈n/2−2⌉-vertex connected, and the bistellar flip graph is (n−3)-vertex connected; both results are tight. The latter bound matches the situation for the subfamily of regular triangulations (i.e., partial triangulations obtained by lifting the points to 3-space and projecting back the lower convex hull), where (n−3)-vertex connectivity has been known since the late eighties through the secondary polytope due to Gelfand, Kapranov, & Zelevinsky and Balinski’s Theorem. For the edge flip-graph, we additionally show that the vertex connectivity is at least as large as (and hence equal to) the minimum degree (i.e., the minimum number of flippable edges in any full triangulation), provided that n is large enough. Our methods also yield several other results: (i) The edge flip graph can be covered by graphs of polytopes of dimension ⌈n/2−2⌉ (products of associahedra) and 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 of a point set are regular iff the partial order of partial subdivisions has height n−3. (iv) There are arbitrarily large sets P with non-regular partial triangulations and such that every proper subset has only regular triangulations, i.e., there are no small certificates for the existence of non-regular triangulations. AU - Wagner, Uli AU - Welzl, Emo ID - 12129 IS - 4 JF - Discrete & Computational Geometry KW - Computational Theory and Mathematics KW - Discrete Mathematics and Combinatorics KW - Geometry and Topology KW - Theoretical Computer Science SN - 0179-5376 TI - Connectivity of triangulation flip graphs in the plane VL - 68 ER - TY - JOUR AB - We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 12148 JF - Forum of Mathematics, Sigma KW - Computational Mathematics KW - Discrete Mathematics and Combinatorics KW - Geometry and Topology KW - Mathematical Physics KW - Statistics and Probability KW - Algebra and Number Theory KW - Theoretical Computer Science KW - Analysis SN - 2050-5094 TI - Rank-uniform local law for Wigner matrices VL - 10 ER - TY - JOUR AB - Inspired by the study of loose cycles in hypergraphs, we define the loose core in hypergraphs as a structurewhich mirrors the close relationship between cycles and $2$-cores in graphs. We prove that in the $r$-uniform binomial random hypergraph $H^r(n,p)$, the order of the loose core undergoes a phase transition at a certain critical threshold and determine this order, as well as the number of edges, asymptotically in the subcritical and supercritical regimes. Our main tool is an algorithm called CoreConstruct, which enables us to analyse a peeling process for the loose core. By analysing this algorithm we determine the asymptotic degree distribution of vertices in the loose core and in particular how many vertices and edges the loose core contains. As a corollary we obtain an improved upper bound on the length of the longest loose cycle in $H^r(n,p)$. AU - Cooley, Oliver AU - Kang, Mihyun AU - Zalla, Julian ID - 12286 IS - 4 JF - The Electronic Journal of Combinatorics KW - Computational Theory and Mathematics KW - Geometry and Topology KW - Theoretical Computer Science KW - Applied Mathematics KW - Discrete Mathematics and Combinatorics TI - Loose cores and cycles in random hypergraphs VL - 29 ER - TY - JOUR AB - Suppose that n is not a prime power and not twice a prime power. We prove that for any Hausdorff compactum X with a free action of the symmetric group Sn, there exists an Sn-equivariant map X→Rn whose image avoids the diagonal {(x,x,…,x)∈Rn∣x∈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 Sn-equivariant maps from the boundary ∂Δn−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 - 11446 IS - 3 JF - Discrete & Computational Geometry KW - Computational Theory and Mathematics KW - Discrete Mathematics and Combinatorics KW - Geometry and Topology KW - Theoretical Computer Science SN - 0179-5376 TI - Vanishing of all equivariant obstructions and the mapping degree VL - 66 ER - TY - JOUR AB - We quantise Whitney’s construction to prove the existence of a triangulation for any C^2 manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric. AU - Boissonnat, Jean-Daniel AU - Kachanovich, Siargey AU - Wintraecken, Mathijs ID - 8940 IS - 1 JF - Discrete & Computational Geometry KW - Theoretical Computer Science KW - Computational Theory and Mathematics KW - Geometry and Topology KW - Discrete Mathematics and Combinatorics SN - 0179-5376 TI - Triangulating submanifolds: An elementary and quantified version of Whitney’s method VL - 66 ER - TY - JOUR AB - The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weight sum of the cut edges. Here, we introduce a linear-time algorithm to compute near-minimum cuts. Our algorithm is based on cluster contraction using label propagation and Padberg and Rinaldi’s contraction heuristics [SIAM Review, 1991]. We give both sequential and shared-memory parallel implementations of our algorithm. Extensive experiments on both real-world and generated instances show that our algorithm finds the optimal cut on nearly all instances significantly faster than other state-of-the-art exact algorithms, and our error rate is lower than that of other heuristic algorithms. In addition, our parallel algorithm runs a factor 7.5× faster on average when using 32 threads. To further speed up computations, we also give a version of our algorithm that performs random edge contractions as preprocessing. This version achieves a lower running time and better parallel scalability at the expense of a higher error rate. AU - Henzinger, Monika H AU - Noe, Alexander AU - Schulz, Christian AU - Strash, Darren ID - 11657 JF - ACM Journal of Experimental Algorithmics KW - Theoretical Computer Science SN - 1084-6654 TI - Practical minimum cut algorithms VL - 23 ER - TY - JOUR AB - The goal of this paper is to present to nonspecialists what is perhaps the simplest possible geometrical picture explaining the mechanism of Arnold diffusion. We choose to speak of a specific model—that of geometric rays in a periodic optical medium. This model is equivalent to that of a particle in a periodic potential in ${\mathbb R}^{n}$ with energy prescribed and to the geodesic flow in a Riemannian metric on ${\mathbb R}^{n} $. AU - Kaloshin, Vadim AU - Levi, Mark ID - 8509 IS - 4 JF - SIAM Review KW - Theoretical Computer Science KW - Applied Mathematics KW - Computational Mathematics SN - 0036-1445 TI - Geometry of Arnold diffusion VL - 50 ER -