# A Fixed-Parameter Perspective on #BIS

## TL;DR

This paper explores the parameterized complexity of counting independent sets in bipartite graphs, revealing NP-hardness of approximation and fixed-parameter tractability in bounded degree graphs, thus mapping the complexity landscape of these problems.

## Contribution

It provides a comprehensive complexity landscape for variants of #BIS parameterized by independent set size, including hardness results and fixed-parameter algorithms.

## Key findings

- NP-hard to approximate within any polynomial ratio
- First problem admits an FPTRAS despite W[1]-hardness
- All problems are fixed-parameter tractable on bounded degree graphs

## Abstract

The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class #RH\Pi_1. It is believed that #BIS does not have an efficient approximation algorithm but also that it is not NP-hard. We study the robustness of the intermediate complexity of #BIS by considering variants of the problem parameterised by the size of the independent set. We exhaustively map the complexity landscape for three problems, with respect to exact computation and approximation and with respect to conventional and parameterised complexity. The three problems are counting independent sets of a given size, counting independent sets with a given number of vertices in one vertex class and counting maximum independent sets amongst those with a given number of vertices in one vertex class. Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio. (This is surprising because the corresponding problems without the size parameter are complete in #RH\Pi_1, and hence are not believed to be NP-hard.) We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate even within any polynomial ratio. Finally, we show that, when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms.