1,927 research outputs found

    Algorithms and complexity for approximately counting hypergraph colourings and related problems

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    The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Δ, this problem falls into the regime of Lovász local lemma when Δ ≲ qᵏ. In prior, however, fast approximate counting algorithms exist when Δ ≲ qᵏ/³, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows. • When q, k ≥ 4 are evens and Δ ≥ 5·qᵏ/², approximating the number of hypergraph colourings is NP-hard. • When the input hypergraph is linear and Δ ≲ qᵏ/², a fast approximate counting algorithm does exist

    Categorical Invariants of Graphs and Matroids

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    Graphs and matroids are two of the most important objects in combinatorics.We study invariants of graphs and matroids that behave well with respect to certain morphisms by realizing these invariants as functors from a category of graphs (resp. matroids). For graphs, we study invariants that respect deletions and contractions ofedges. For an integer g>0g > 0, we define a category of Ggop\mathcal{G}^{op}_g of graphs of genus at most g where morphisms correspond to deletions and contractions. We prove that this category is locally Noetherian and show that many graph invariants form finitely generated modules over the category Ggop\mathcal{G}^{op}_g. This fact allows us to exihibit many stabilization properties of these invariants. In particular we show that the torsion that can occur in the homologies of the unordered configuration space of n points in a graph and the matching complex of a graph are uniform over the entire family of graphs with genus gg. For matroids, we study valuative invariants of matroids. Given a matroid,one can define a corresponding polytope called the base polytope. Often, the base polytope of a matroid can be decomposed into a cell complex made up of base polytopes of other matroids. A valuative invariant of matroids is an invariant that respects these polytope decompositions. We define a category Mid\mathcal{M}^{\wedge}_{id} of matroids whose morphisms correspond to containment of base polytopes. We then define the notion of a categorical matroid invariant which categorifies the notion of a valuative invariant. Finally, we prove that the functor sending a matroid to its Orlik-Solomon algebra is a categorical valuative invariant. This allows us to derive relations among the Orlik-Solomon algebras of a matroid and matroids that decompose its base polytope viewed as representations of any group Γ\Gamma whose action is compatible with the polytope decomposition. This dissertation includes previously unpublished co-authored material

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Connectivity gaps among matroids with the same enumerative invariants

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    Many important enumerative invariants of a matroid can be obtained from its Tutte polynomial, and many more are determined by two stronger invariants, the G\mathcal{G}-invariant and the configuration of the matroid. We show that the same is not true of the most basic connectivity invariants. Specifically, we show that for any positive integer nn, there are pairs of matroids that have the same configuration (and so the same G\mathcal{G}-invariant and the same Tutte polynomial) but the difference between their Tutte connectivities exceeds nn, and likewise for vertical connectivity and branch-width. The examples that we use to show this, which we construct using an operation that we introduce, are transversal matroids that are also positroids

    Dynamic Pricing Schemes in Combinatorial Markets

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    In combinatorial markets where buyers are self-interested, the buyers may make purchases that lead to suboptimal item allocations. As a central coordinator, our goal is to impose prices on the items of the market so that its buyers are incentivized to exclusively make optimal purchases. In this thesis, we study the question of whether dynamic pricing schemes can achieve the optimal social welfare in multi-demand combinatorial markets. This well-motivated question has been the topic of some study, but has remained mostly open, and to date, positive results are only known for extremal cases. In this thesis, we present the current results for unit-demand, bi-demand and tri-demand markets. In the context of these results, we discuss the significance of not having a deficiency of items, which is known as the (OPT) condition. We outline an approach for handling an item deficiency, and we expose barriers to extending the known techniques to markets of larger demand

    Selfadhesivity in Gaussian conditional independence structures

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    Selfadhesivity is a property of entropic polymatroids which guarantees that the polymatroid can be glued to an identical copy of itself along arbitrary restrictions such that the two pieces are independent given the common restriction. We show that positive definite matrices satisfy this condition as well and examine consequences for Gaussian conditional independence structures. New axioms of Gaussian CI are obtained by applying selfadhesivity to the previously known axioms of structural semigraphoids and orientable gaussoids.Comment: 13 pages; v3: minor revisio

    Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices

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    We address the computation of the degrees of minors of a noncommutative symbolic matrix of form A[c]:=k=1mAktckxk, A[c] := \sum_{k=1}^m A_k t^{c_k} x_k, where AkA_k are matrices over a field K\mathbb{K}, xix_i are noncommutative variables, ckc_k are integer weights, and tt is a commuting variable specifying the degree. This problem extends noncommutative Edmonds' problem (Ivanyos et al. 2017), and can formulate various combinatorial optimization problems. Extending the study by Hirai 2018, and Hirai, Ikeda 2022, we provide novel duality theorems and polyhedral characterization for the maximum degrees of minors of A[c]A[c] of all sizes, and develop a strongly polynomial-time algorithm for computing them. This algorithm is viewed as a unified algebraization of the classical Hungarian method for bipartite matching and the weight-splitting algorithm for linear matroid intersection. As applications, we provide polynomial-time algorithms for weighted fractional linear matroid matching and linear optimization over rank-2 Brascamp-Lieb polytopes

    Polyhedral Geometry in OSCAR

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    OSCAR is an innovative new computer algebra system which combines and extends the power of its four cornerstone systems - GAP (group theory), Singular (algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic (number theory). Here, we give an introduction to polyhedral geometry computations in OSCAR, as a chapter of the upcoming OSCAR book. In particular, we define polytopes, polyhedra, and polyhedral fans, and we give a brief overview about computing convex hulls and solving linear programs. Three detailed case studies are concerned with face numbers of random polytopes, constructions and properties of Gelfand-Tsetlin polytopes, and secondary polytopes.Comment: 19 pages, 8 figure

    Inequalities for totally nonnegative matrices: Gantmacher--Krein, Karlin, and Laplace

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    A real linear combination of products of minors which is nonnegative over all totally nonnegative (TN) matrices is called a determinantal inequality for these matrices. It is referred to as multiplicative when it compares two collections of products of minors and additive otherwise. Set theoretic operations preserving the class of TN matrices naturally translate into operations preserving determinantal inequalities in this class. We introduce set row/column operations that act directly on all determinantal inequalities for TN matrices, and yield further inequalities for these matrices. These operations assist in revealing novel additive inequalities for TN matrices embedded in the classical identities due to Laplace [[Mem.. Acad.. Sciences Paris 1772]1772] and Karlin (1968).(1968). In particular, for any square TN matrix A,A, these derived inequalities generalize -- to every i^{\mbox{th}} row of AA and j^{\mbox{th}} column of adjA{\rm adj} A -- the classical Gantmacher--Krein fluctuating inequalities (1941)(1941) for i=j=1.i=j=1. Furthermore, our row/column operations reveal additional undiscovered fluctuating inequalities for TN matrices. The introduced set row/column operations naturally birth an algorithm that can detect certain determinantal expressions that do not form an inequality for TN matrices. However, the algorithm completely characterizes the multiplicative inequalities comparing products of pairs of minors. Moreover, the underlying row/column operations add that these inequalities are offshoots of certain ''complementary/higher'' ones. These novel results seem very natural, and in addition thoroughly describe and enrich the classification of these multiplicative inequalities due to Fallat--Gekhtman--Johnson [[Adv.. Appl.. Math.. 2003]2003] and later Skandera [[J.. Algebraic Comb.. 2004].2004].Comment: 2 figures, 39 pages, and minor adjustments in the expositio

    DD-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals

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    Feynman integrals are solutions to linear partial differential equations with polynomial coefficients. Using a triangle integral with general exponents as a case in point, we compare DD-module methods to dedicated methods developed for solving differential equations appearing in the context of Feynman integrals, and provide a dictionary of the relevant concepts. In particular, we implement an algorithm due to Saito, Sturmfels, and Takayama to derive canonical series solutions of regular holonomic DD-ideals, and compare them to asymptotic series derived by the respective Fuchsian systems.Comment: 35 pages, 2 figures, 2 appendices; comments welcom
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