Complexity of Maximum Fixed Point Problem 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: component i has a positive (resp. negative) influence on component j meaning that j tends to mimic (resp. negate) i. The digraph of influences is called signed interaction digraph (SID), and one SID may correspond to a large number of BNs (which is, in average, doubly exponential according to n). 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, NP NP , NP #P or NEXPTIME. In particular, we prove that it is NP-complete (resp. NEXPTIME-complete) to decide if a given SID can correspond to a BN having at least two fixed points (resp. no fixed point).

Complexity of limit-cycle problems in Boolean networks

“…Finally, if k is part of the input, is there a drastic complexity increase as observed for problems related to the number of fixed points in [6]? The construction presented in the proof of Theorem 4 makes heavy use of being a k constant.…”

“…Eventually, questions on the maximum number of fixed points possible when only the interaction digraph of a BN is provided, have already let some complexity classes higher than NP appear in problems related to the attractors of BNs [6].…”

“…In relation to complexity theory, the main known results based on BNs are: determining if a BN admits fixed points is NP-complete, counting fixed points is #P-complete [9,19], determining if a fixed point has a non-trivial attraction basin is NP-complete, determining if there exists another update schedule that conserves the limit-cycles of a given BN evolving in parallel is NP-hard [4]. Moreover, a recent work [6] focused on related questions on fixed point complexity by focusing on interaction digraphs and not on BNs anymore (notice that several BNs admit the same interaction digraph).…”

Counting Periodic Points in Parallel Graph Dynamical Systems

“…In this paper, we use such maximal independent sets to establish the minimum number of 2-periodic points in homogeneous PDS induced by a maxterm or minterm. Regarding the minimum number of fixed points' problem, in the recent work [48], the authors studied its complexity and established that finding a Boolean network having at least k fixed points is in P or complete for NP, NP # P, or NEXPTIME, depending on the input.…”

Optimising attractor computation in Boolean automata networks

“…The dynamics of a BAN is usually partitionned in two sorts of configurations: the recurring ones that are parts of attractors and either belong to a limit cycle or are fixed points; the others that evolve towards these attractors and belong to their attraction basins. The questions of characterising, computing or counting those attractors from a simple description of the network have been explored [8,1,9,7,14,2], and has been shown to be difficult problems [8,16,3,4,15]. In this paper, we propose a new method for computing the attractors of a BAN under the parallel update schedule.…”