A Fixed-Parameter Perspective on #BIS

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 #RHΠ 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 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 #RHΠ 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.

Counting independent sets in cocomparability graphs

“…Proof. By the same argument as Theorem 6, using that it is #W[1]-complete to count independent sets of size k in bipartite graphs [3,Theorem 4]. The following are proved similarly to the correponding results above.…”

Computing the Number of Induced Copies of a Fixed Graph in a Bounded Degree Graph

“…(Here thẽ O-notation means that we suppress polynomial factors in m.) Remark 1.1. Theorem 13 in [5] in fact concerns vertex-coloured graphs H and G. Our proof of Theorem 1.1 also easily extends to the coloured setting. We discuss this in Section 4.…”

“…However, when the graph G is of bounded degree, which is often of interest in statistical physics, the problem is no longer W [ Goldberg, and Lapinskas [5,Theorem 13] showed that for a graph H on m vertices and a bounded degree graph G on n vertices, ind(H, G) can be computed in time O(nm O(m) ), thus giving an FPT algorithm in terms of m. In the present paper we go further and give an algorithm with essentially optimal running time. We assume the standard word-RAM machine model with logarithmic-sized words.…”

Stable Matchings with Restricted Preferences: Structure and Complexity

“…Interestingly, #BIS shows a similar threshold around = 6: counting independent sets in bipartite graphs of maximum degree ≥ 6 is #BIS-complete [4], while instances of maximum degree ≤ 5 admit an FPRAS, even if vertices on only one side of the bipartition obey the degree bound [26]. Recently, Curticapean et al [11] considered other parameterized variants of #BIS, and showed that the variants admit efficient fixed-parameter algorithms in bounded-degree graphs.…”

“…Finally, given a color ∈ , we denote its "next" and "previous" colors in by + and − , respectively. 11 That is, if = ( ), then + = ( + 1) and − = ( − 1). Lemma 3.2.…”