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 λc=k(d−1)d+1dd
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<λc, there is an FPTAS for computing the partition
function; and when λ=λc, 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, 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