Partition functions for dense instances of combinatorial enumeration problems

TL;DR

This paper studies partition functions in weighted complete graphs, showing concentration on simple structures under certain weight conditions, enabling efficient algorithms to distinguish graphs with many Hamiltonian cycles.

Contribution

It introduces a novel analysis of partition functions for dense combinatorial problems, leading to polynomial-time algorithms for graph property testing.

Findings

Partition functions concentrate on simple structures when edge weights are similar.

Efficient algorithms can distinguish graphs with many Hamiltonian cycles.

The approach applies to cycle covers, walks, and spanning trees.

Abstract

Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds: cycle covers, closed walks, spanning trees. We show that if the weights of the edges of the graph are within a constant factor, fixed in advance, of each other then the bulk of the partition function is concentrated on the subsets of a particularly simple structure: cycle covers with few cycles, walks that visit every vertex only few times, and spanning trees with small degree of every vertex. This allows us to construct a polynomial time algorithm to separate graphs with many Hamiltonian cycles from graphs that are sufficiently far from Hamiltonian.