Generalising the maximum independent set algorithm via Boolean networks

Abstract

A simple greedy algorithm to find a maximal independent set (MIS) in a graph starts with the empty set and visits every vertex, adding it to the set if and only if none of its neighbours are already in the set. In this paper, we consider (the complexity of decision problems related to) the generalisation of this MIS algorithm wherein any starting set is allowed. Two main approaches are leveraged. Firstly, we view the MIS algorithm as a sequential update of a Boolean network according to a permutation of the vertex set. Secondly, we introduce the concept of a constituency of a graph: a set of vertices that is dominated by an independent set. Recognizing a constituency is NP-complete, a fact we leverage repeatedly in our investigation.

Our contributions are multiple: we establish that deciding whether all maximal independent sets can be reached from some configuration is coNP-complete; that fixing words (which reach a MIS from any starting configuration) and fixing permutations (briefly, permises) are coNP-complete to recognize; and that permissible graphs (graphs with a permis) are coNP-hard to recognize. We also exhibit large classes of permissible and non-permissible graphs, notably near-comparability graphs which may be of independent interest.

Lastly, we extend our study to digraphs, where we search for kernels. Since the natural generalisation of our approach may not necessarily find a kernel, we introduce two further Boolean networks for digraphs: one always finds an independent set, and the other always finds a dominating set.