436 research outputs found

    Counting hypergraph matchings up to uniqueness threshold

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    We study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges. With an activity parameter λ\lambda, each matching MM is assigned a weight λM\lambda^{|M|}. The counting problem is formulated as computing a partition function that gives the sum of the weights of all matchings in a hypergraph. This problem unifies two extensively studied statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer-dimer model (graph matchings). For this model, the critical activity λc=ddk(d1)d+1\lambda_c= \frac{d^d}{k (d-1)^{d+1}} is the threshold for the uniqueness of Gibbs measures on the infinite (d+1)(d+1)-uniform (k+1)(k+1)-regular hypertree. Consider hypergraphs of maximum degree at most k+1k+1 and maximum size of hyperedges at most d+1d+1. We show that when λ<λc\lambda < \lambda_c, there is an FPTAS for computing the partition function; and when λ=λc\lambda = \lambda_c, there is a PTAS for computing the log-partition function. These algorithms are based on the decay of correlation (strong spatial mixing) property of Gibbs distributions. When λ>2λc\lambda > 2\lambda_c, there is no PRAS for the partition function or the log-partition function unless NP==RP. Towards obtaining a sharp transition of computational complexity of approximate counting, we study the local convergence from a sequence of finite hypergraphs to the infinite lattice with specified symmetry. We show a surprising connection between the local convergence and the reversibility of a natural random walk. This leads us to a barrier for the hardness result: The non-uniqueness of infinite Gibbs measure is not realizable by any finite gadgets

    Most primitive groups are full automorphism groups of edge-transitive hypergraphs

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    We prove that, for a primitive permutation group G acting on a set of size n, other than the alternating group, the probability that Aut(X,Y^G) = G for a random subset Y of X, tends to 1 as n tends to infinity. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M. Klin. Moreover, we give an upper bound n^{1/2+\epsilon} for the minimum size of the edges in such a hypergraph. This is essentially best possible.Comment: To appear in special issue of Journal of Algebra in memory of Akos Seres

    Asymptotics for incidence matrix classes

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    We define {\em incidence matrices} to be zero-one matrices with no zero rows or columns. A classification of incidence matrices is considered for which conditions of symmetry by transposition, having no repeated rows/columns, or identification by permutation of rows/columns are imposed. We find asymptotics and relationships for the number of matrices with nn ones in these classes as nn\to\infty.Comment: updated and slightly expanded versio

    Generalized Kneser coloring theorems with combinatorial proofs

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    The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol'nikov, Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver's theorem.Comment: 19 pages, 4 figure

    Asymptotic enumeration of incidence matrices

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    We discuss the problem of counting {\em incidence matrices}, i.e. zero-one matrices with no zero rows or columns. Using different approaches we give three different proofs for the leading asymptotics for the number of matrices with nn ones as nn\to\infty. We also give refined results for the asymptotic number of i×ji\times j incidence matrices with nn ones.Comment: jpconf style files. Presented at the conference "Counting Complexity: An international workshop on statistical mechanics and combinatorics." In celebration of Prof. Tony Guttmann's 60th birthda
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