FPTAS for #BIS with Degree Bounds on One Side

TL;DR

This paper presents a simple deterministic FPTAS for counting independent sets in bipartite graphs when one side's maximum degree is at most 5, expanding the class of tractable instances.

Contribution

It introduces a novel FPTAS for #BIS with degree bounds on one side, where previously such schemes required bounds on both sides.

Findings

FPTAS exists for #BIS with one side degree ≤ 5

No degree bound needed on the other side

Simplifies previous approaches for approximate counting

Abstract

Counting the number of independent sets for a bipartite graph (#BIS) plays a crucial role in the study of approximate counting. It has been conjectured that there is no fully polynomial-time (randomized) approximation scheme (FPTAS/FPRAS) for #BIS, and it was proved that the problem for instances with a maximum degree of $6$ is already as hard as the general problem. In this paper, we obtain a surprising tractability result for a family of #BIS instances. We design a very simple deterministic fully polynomial-time approximation scheme (FPTAS) for #BIS when the maximum degree for one side is no larger than $5$. There is no restriction for the degrees on the other side, which do not even have to be bounded by a constant. Previously, FPTAS was only known for instances with a maximum degree of $5$ for both sides.