Counting Weighted Independent Sets beyond the Permanent

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