1,569 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Quantum ergodicity on the Bruhat-Tits building for PGL(3,F)\text{PGL}(3, F) in the Benjamini-Schramm limit

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    We study eigenfunctions of the spherical Hecke algebra acting on L2(Γn\G/K)L^2(\Gamma_n \backslash G / K) where G=PGL(3,F)G = \text{PGL}(3, F) with FF a non-archimedean local field of characteristic zero, K=PGL(3,O)K = \text{PGL}(3, \mathcal{O}) with O\mathcal{O} the ring of integers of FF, and (Γn)(\Gamma_n) is a sequence of cocompact torsionfree lattices. We prove a form of equidistribution on average for eigenfunctions whose spectral parameters lie in the tempered spectrum when the associated sequence of quotients of the Bruhat-Tits building Benjamini-Schramm converges to the building itself.Comment: 111 pages, 25 figures, 2 table

    Negative definite spin filling and branched double covers

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    We investigate the negative definite spin fillings of branched double covers of alternating knots. We derive some obstructions for the existence of such fillings and find a characterization of special alternating knots based on them.Comment: 22 pages, 16 figure

    The Complexity of Recognizing Geometric Hypergraphs

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    As set systems, hypergraphs are omnipresent and have various representations ranging from Euler and Venn diagrams to contact representations. In a geometric representation of a hypergraph H=(V,E)H=(V,E), each vertex vVv\in V is associated with a point pvRdp_v\in \mathbb{R}^d and each hyperedge eEe\in E is associated with a connected set seRds_e\subset \mathbb{R}^d such that {pvvV}se={pvve}\{p_v\mid v\in V\}\cap s_e=\{p_v\mid v\in e\} for all eEe\in E. We say that a given hypergraph HH is representable by some (infinite) family FF of sets in Rd\mathbb{R}^d, if there exist PRdP\subset \mathbb{R}^d and SFS \subseteq F such that (P,S)(P,S) is a geometric representation of HH. For a family F, we define RECOGNITION(F) as the problem to determine if a given hypergraph is representable by F. It is known that the RECOGNITION problem is R\exists\mathbb{R}-hard for halfspaces in Rd\mathbb{R}^d. We study the families of translates of balls and ellipsoids in Rd\mathbb{R}^d, as well as of other convex sets, and show that their RECOGNITION problems are also R\exists\mathbb{R}-complete. This means that these recognition problems are equivalent to deciding whether a multivariate system of polynomial equations with integer coefficients has a real solution.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023) 17 pages, 11 figure

    The Distributed Complexity of Locally Checkable Labeling Problems Beyond Paths and Trees

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    We consider locally checkable labeling LCL problems in the LOCAL model of distributed computing. Since 2016, there has been a substantial body of work examining the possible complexities of LCL problems. For example, it has been established that there are no LCL problems exhibiting deterministic complexities falling between ω(logn)\omega(\log^* n) and o(logn)o(\log n). This line of inquiry has yielded a wealth of algorithmic techniques and insights that are useful for algorithm designers. While the complexity landscape of LCL problems on general graphs, trees, and paths is now well understood, graph classes beyond these three cases remain largely unexplored. Indeed, recent research trends have shifted towards a fine-grained study of special instances within the domains of paths and trees. In this paper, we generalize the line of research on characterizing the complexity landscape of LCL problems to a much broader range of graph classes. We propose a conjecture that characterizes the complexity landscape of LCL problems for an arbitrary class of graphs that is closed under minors, and we prove a part of the conjecture. Some highlights of our findings are as follows. 1. We establish a simple characterization of the minor-closed graph classes sharing the same deterministic complexity landscape as paths, where O(1)O(1), Θ(logn)\Theta(\log^* n), and Θ(n)\Theta(n) are the only possible complexity classes. 2. It is natural to conjecture that any minor-closed graph class shares the same complexity landscape as trees if and only if the graph class has bounded treewidth and unbounded pathwidth. We prove the "only if" part of the conjecture. 3. In addition to the well-known complexity landscapes for paths, trees, and general graphs, there are infinitely many different complexity landscapes among minor-closed graph classes

    Excluding Surfaces as Minors in Graphs

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    We introduce an annotated extension of treewidth that measures the contribution of a vertex set XX to the treewidth of a graph G.G. This notion provides a graph distance measure to some graph property P\mathcal{P}: A vertex set XX is a kk-treewidth modulator of GG to P\mathcal{P} if the treewidth of XX in GG is at most kk and its removal gives a graph in P.\mathcal{P}.This notion allows for a version of the Graph Minors Structure Theorem (GMST) that has no need for apices and vortices: KkK_k-minor free graphs are those that admit tree-decompositions whose torsos have ckc_{k}-treewidth modulators to some surface of Euler-genus ck.c_{k}. This reveals that minor-exclusion is essentially tree-decomposability to a ``modulator-target scheme'' where the modulator is measured by its treewidth and the target is surface embeddability. We then fix the target condition by demanding that Σ\Sigma is some particular surface and define a ``surface extension'' of treewidth, where \Sigma\mbox{-}\mathsf{tw}(G) is the minimum kk for which GG admits a tree-decomposition whose torsos have a kk-treewidth modulator to being embeddable in Σ.\Sigma.We identify a finite collection DΣ\mathfrak{D}_{\Sigma} of parametric graphs and prove that the minor-exclusion of the graphs in DΣ\mathfrak{D}_{\Sigma} precisely determines the asymptotic behavior of {\Sigma}\mbox{-}\mathsf{tw}, for every surface Σ.\Sigma. It follows that the collection DΣ\mathfrak{D}_{\Sigma} bijectively corresponds to the ``surface obstructions'' for Σ,\Sigma, i.e., surfaces that are minimally non-contained in $\Sigma.

    Planar Disjoint Paths, Treewidth, and Kernels

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    In the Planar Disjoint Paths problem, one is given an undirected planar graph with a set of kk vertex pairs (si,ti)(s_i,t_i) and the task is to find kk pairwise vertex-disjoint paths such that the ii-th path connects sis_i to tit_i. We study the problem through the lens of kernelization, aiming at efficiently reducing the input size in terms of a parameter. We show that Planar Disjoint Paths does not admit a polynomial kernel when parameterized by kk unless coNP \subseteq NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e}, Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel unless the WK-hierarchy collapses. Our reduction carries over to the setting of edge-disjoint paths, where the kernelization status remained open even in general graphs. On the positive side, we present a polynomial kernel for Planar Disjoint Paths parameterized by k+twk + tw, where twtw denotes the treewidth of the input graph. As a consequence of both our results, we rule out the possibility of a polynomial-time (Turing) treewidth reduction to tw=kO(1)tw= k^{O(1)} under the same assumptions. To the best of our knowledge, this is the first hardness result of this kind. Finally, combining our kernel with the known techniques [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver, SICOMP'94] yields an alternative (and arguably simpler) proof that Planar Disjoint Paths can be solved in time 2O(k2)nO(1)2^{O(k^2)}\cdot n^{O(1)}, matching the result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].Comment: To appear at FOCS'23, 82 pages, 30 figure

    Geometry of the doubly periodic Aztec dimer model

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    The purpose of the present work is to provide a detailed asymptotic analysis of the k×k\times\ell doubly periodic Aztec diamond dimer model of growing size for any kk and \ell and under mild conditions on the edge weights. We explicitly describe the limit shape and the 'arctic' curves that separate different phases, as well as prove the convergence of local fluctuations to the appropriate translation-invariant Gibbs measures away from the arctic curves. We also obtain a homeomorphism between the rough region and the amoeba of an associated Harnack curve, and illustrate, using this homeomorphism, how the geometry of the amoeba offers insight into various aspects of the geometry of the arctic curves. In particular, we determine the number of frozen and smooth regions and the number of cusps on the arctic curves. Our framework essentially relies on three somewhat distinct areas: (1) Wiener-Hopf factorization approach to computing dimer correlations; (2) Algebraic geometric `spectral' parameterization of periodic dimer models; and (3) Finite-gap theory of linearization of (nonlinear) integrable partial differential and difference equations on the Jacobians of the associated algebraic curves. In addition, in order to access desired asymptotic results we develop a novel approach to steepest descent analysis on Riemann surfaces via their amoebas.Comment: 94 pages, 22 figure

    Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs

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    For any finite set H = {H1,. .. , Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1,. .. , Hp as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of H-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set

    Size-Ramsey numbers of structurally sparse graphs

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    Size-Ramsey numbers are a central notion in combinatorics and have been widely studied since their introduction by Erd\H{o}s, Faudree, Rousseau and Schelp in 1978. Research has mainly focused on the size-Ramsey numbers of nn-vertex graphs with constant maximum degree Δ\Delta. For example, graphs which also have constant treewidth are known to have linear size-Ramsey numbers. On the other extreme, the canonical examples of graphs of unbounded treewidth are the grid graphs, for which the best known bound has only very recently been improved from O(n3/2)O(n^{3/2}) to O(n5/4)O(n^{5/4}) by Conlon, Nenadov and Truji\'c. In this paper, we prove a common generalization of these results by establishing new bounds on the size-Ramsey numbers in terms of treewidth (which may grow as a function of nn). As a special case, this yields a bound of O~(n3/21/2Δ)\tilde{O}(n^{3/2 - 1/2\Delta}) for proper minor-closed classes of graphs. In particular, this bound applies to planar graphs, addressing a question of Wood. Our proof combines methods from structural graph theory and classic Ramsey-theoretic embedding techniques, taking advantage of the product structure exhibited by graphs with bounded treewidth.Comment: 21 page
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