The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs, motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question. Recently, we were able to establish the trichotomy [arXiv:1312.4524]. Here, we consider connectivity issues of satisfiability problems defined by Boolean circuits and propositional formulas that use gates, resp. connectives, from a fixed set of Boolean functions. We obtain dichotomies for the diameter and the two connectivity problems: on one side, the diameter is linear in the number of variables, and both problems are in P, while on the other side, the diameter can be exponential, and the problems are PSPACE-complete. For partially quantified formulas, we show an analogous dichotomy.Comment: 20 pages, several improvement

Simplest random K-satisfiability problem

We study a simple and exactly solvable model for the generation of random satisfiability problems. These consist of $\gamma N$ random boolean constraints which are to be satisfied simultaneously by $N$ logical variables. In statistical-mechanics language, the considered model can be seen as a diluted p-spin model at zero temperature. While such problems become extraordinarily hard to solve by local search methods in a large region of the parameter space, still at least one solution may be superimposed by construction. The statistical properties of the model can be studied exactly by the replica method and each single instance can be analyzed in polynomial time by a simple global solution method. The geometrical/topological structures responsible for dynamic and static phase transitions as well as for the onset of computational complexity in local search method are thoroughly analyzed. Numerical analysis on very large samples allows for a precise characterization of the critical scaling behaviour.Comment: 14 pages, 5 figures, to appear in Phys. Rev. E (Feb 2001). v2: minor errors and references correcte

Schaefer's theorem for graphs

Schaefer's theorem is a complexity classification result for so-called Boolean constraint satisfaction problems: it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time, or is NP-complete. We present an analog of this dichotomy result for the propositional logic of graphs instead of Boolean logic. In this generalization of Schaefer's result, the input consists of a set W of variables and a conjunction \Phi\ of statements ("constraints") about these variables in the language of graphs, where each statement is taken from a fixed finite set \Psi\ of allowed quantifier-free first-order formulas; the question is whether \Phi\ is satisfiable in a graph. We prove that either \Psi\ is contained in one out of 17 classes of graph formulas and the corresponding problem can be solved in polynomial time, or the problem is NP-complete. This is achieved by a universal-algebraic approach, which in turn allows us to use structural Ramsey theory. To apply the universal-algebraic approach, we formulate the computational problems under consideration as constraint satisfaction problems (CSPs) whose templates are first-order definable in the countably infinite random graph. Our method to classify the computational complexity of those CSPs is based on a Ramsey-theoretic analysis of functions acting on the random graph, and we develop general tools suitable for such an analysis which are of independent mathematical interest.Comment: 54 page

Percolation of satisfiability in finite dimensions

The satisfiability and optimization of finite-dimensional Boolean formulas are studied using percolation theory, rare region arguments, and boundary effects. In contrast with mean-field results, there is no satisfiability transition, though there is a logical connectivity transition. In part of the disconnected phase, rare regions lead to a divergent running time for optimization algorithms. The thermodynamic ground state for the NP-hard two-dimensional maximum-satisfiability problem is typically unique. These results have implications for the computational study of disordered materials.Comment: 4 pages, 4 fig