Counting weighted independent sets beyond the permanent

TL;DR

This paper extends polynomial-time approximation methods for the permanent to broader classes of graphs, specifically (claw, odd hole)-free and (fork, odd hole)-free graphs, using advanced graph decomposition techniques.

Contribution

It generalizes the approximation of the permanent to new graph classes beyond bipartite line graphs, employing structural graph theory methods.

Findings

Extension to (claw, odd hole)-free graphs.

Further extension to (fork, odd hole)-free graphs.

Utilization of graph decompositions and structural results.

Abstract

Jerrum, Sinclair and Vigoda (2004) showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function of edge-weighted matchings in a bipartite graph. Equivalently, this may be viewed as approximating the partition function of vertex-weighted independent sets in the line graph of a bipartite graph. Line graphs of bipartite graphs are perfect graphs, and are known to be precisely the class of (claw, diamond, odd hole)-free graphs. So how far does the result of Jerrum, Sinclair and Vigoda extend? We first show that it extends to (claw, odd hole)-free graphs, and then show that it extends to the even larger class of (fork, odd hole)-free graphs. Our techniques are based on graph decompositions, which have been the focus of much recent work in structural graph theory, and on structural results of Chvatal and Sbihi (1988), Maffray and Reed (1999) and Lozin and Milanic (2008).