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 mathsf {#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 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 mathsf {#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.
Highlights
The problem of counting the independent sets of a bipartite graph, called #BIS, is one of the most important problems in the field of approximate counting
Many approximate counting problems are equivalent in difficulty to #BIS, including those that arise in spin-system problems [12,14] and in other domains
It is already known that the complexity of #BIS is unchanged when the degree of the input graph is restricted [2] but there is an efficient approximation algorithm when a stronger degree restriction is applied, even to the vertices in just one of the parts of the vertex partition of the bipartite graph [17]
Summary
The problem of (approximately) counting the independent sets of a bipartite graph, called #BIS, is one of the most important problems in the field of approximate counting This problem is known to be complete in the intermediate complexity class #RH 1[8]. The NP-hardness of these approximate counting problems is surprising given that the corresponding problems without the parameter k (that is, the problem #BIS and the problem #Max-BIS, which is the problem of counting the maximum independent sets of a bipartite graph) are both complete in #RH 1, and are not believed to be NP-hard. Note that Theorem 14(i) is implicit in independent work by Patel and Regts [20]
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