Complexity of fixed point counting problems in Boolean Networks
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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).
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