Counting hypergraph matchings up to uniqueness threshold

Renjie Song, Yitong Yin, Jinman Zhao

Abstract: We study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges. With an activity parameter λ, each matching M is assigned a weight λ | 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 problem, the critical activity λ c = d d k ( d − 1 ) d + 1 is the threshold for the uniqueness of Gibbs measures on the infinite ( d + 1 ) -uniform ( k + 1 ) -regular hypertree. Consider hypergraphs of maximum degree at most k + 1 and maximum size of hyperedges at most d + 1 . We show that when λ < λ c , there is an FPTAS for computing the partition function; and when λ = λ 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 > 2 λ 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.