Complexity of limit cycles with block-sequential update schedules in conjunctive networks

TL;DR

This paper investigates the computational complexity of determining limit cycles in conjunctive Boolean networks under various update schedules, establishing NP-completeness results and identifying tractable cases.

Contribution

It proves NP-completeness of the limit cycle decision problem for general and block-sequential schedules, and identifies polynomial cases for strongly connected digraphs.

Findings

NP-complete for general conjunctive networks with variable cycle length

Polynomial-time solvable when the interaction digraph is strongly connected

NP-complete for fixed cycle length with block-sequential schedules

Abstract

In this paper, we deal the following decision problem: given a conjunctive Boolean network defined by its interaction digraph, does it have a limit cycle of a given length k? We prove that this problem is NP-complete in general if k is a parameter of the problem and in P if the interaction digraph is strongly connected. The case where $k$ is a constant, but the interaction digraph is not strongly connected remains open. Furthermore, we study the variation of the decision problem: given a conjunctive Boolean network, does there exist a block-sequential (resp. sequential) update schedule such that there exists a limit cycle of length k? We prove that this problem is NP-complete for any constant k >= 2.