Complexity of fixed point counting problems in Boolean networks

Abstract

A Boolean network (BN) with n components is a discrete dynamical system described by the successive iterations of a function f:{0,1}n→{0,1}n. This model finds applications in biology, where fixed points play a central role. For example, in genetic regulations, they correspond to cell phenotypes. In this context, experiments reveal the existence of positive or negative influences among components. The digraph of influences is called signed interaction digraph (SID), and one SID may correspond to a large number of BNs. The present work opens a new perspective on the well-established study of fixed points in BNs. When biologists discover the SID of a BN they do not know, they may ask: given that SID, can it correspond to a BN having at least/at most k fixed points? Depending on the input, we prove that these problems are in P or complete for NP, NPNP, NP#P or NEXPTIME.