Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

TL;DR

This paper classifies the computational complexity of the fixed-point existence problem in boolean dynamical systems, establishing clear dichotomies between NP-complete and polynomial-time decidable cases based on function and graph classes.

Contribution

It provides a complete classification of fixed-point problems for boolean dynamical systems, identifying precise conditions for NP-completeness and polynomial-time decidability.

Findings

NP-complete when F contains self-dual functions and G contains planar graphs

Decidable in polynomial time otherwise

Dichotomy theorems for various graph and function classes

Abstract

A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph classes G, an (F, G)-system is a boolean dynamical system such that all local transition functions lie in F and the underlying graph lies in G. Let F be a class of boolean functions which is closed under composition and let G be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If F contains the self-dual functions and G contains the planar graphs then the fixed-point existence problem for (F, G)-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If F contains the self-dual functions and G contains the graphs having vertex covers of size one then the fixed-point existence problem for (F, G)-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.