2,292 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 λ∣M∣\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(d−1)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

    On the Chromatic Thresholds of Hypergraphs

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    Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least c(∣V(H)∣r−1)c \binom{|V(H)|}{r-1} has bounded chromatic number. This parameter has a long history for graphs (r=2), and in this paper we begin its systematic study for hypergraphs. {\L}uczak and Thomass\'e recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Tur\'an number is achieved uniquely by the complete (r+1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of nondegenerate hypergraphs whose Tur\'an number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle. In order to prove upper bounds we introduce the concept of fiber bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fiber bundle dimension, a structural property of fiber bundles that is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemer\'edi for graphs and might be of independent interest. Many open problems remain.Comment: 37 pages, 4 figure

    3-uniform hypergraphs: modular decomposition and realization by tournaments

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    Let HH be a 3-uniform hypergraph. A tournament TT defined on V(T)=V(H)V(T)=V(H) is a realization of HH if the edges of HH are exactly the 3-element subsets of V(T)V(T) that induce 3-cycles. We characterize the 3-uniform hypergraphs that admit realizations by using a suitable modular decomposition

    The Ising Partition Function: Zeros and Deterministic Approximation

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    We study the problem of approximating the partition function of the ferromagnetic Ising model in graphs and hypergraphs. Our first result is a deterministic approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters β\beta (the interaction) and λ\lambda (the external field), except for the case ∣λ∣=1\vert{\lambda}\vert=1 (the "zero-field" case). A randomized algorithm (FPRAS) for all graphs, and all β,λ\beta,\lambda, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the "decay of correlations" property. Rather, we exploit and extend machinery developed recently by Barvinok, and Patel and Regts, based on the location of the complex zeros of the partition function, which can be seen as an algorithmic realization of the classical Lee-Yang approach to phase transitions. Our approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this extension, we establish a tight version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.Comment: clarified presentation of combinatorial arguments, added new results on optimality of univariate Lee-Yang theorem
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