Year: 2015
Type: preprint
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 model, the critical activity $λ_c= \frac{d^d}{k (d-1)^{d+1}}$ 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 $λ> 2λ_c$, 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.
Source: arXiv (Cornell University)
Authors: Song, Renjie, Yin, Yitong, Zhao, Jinman
Institutions: Nanjing University, University of Wisconsin–Madison
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G4430459587, G6667581590
OpenAlex ID: W2133453606