A linear-time transformation of linear inequalities into conjunctive normal form

We present a technique that transforms any binary programming problem with integral coefficients to a satisfiability problem of propositional logic in linear time. Preliminary computational experience using this transformation, shows that a pure logical solver can be a valuable tool for solving binary programming problems. In a number of cases it competes favourably with well known techniques from operations research, especially for hard unsatisfiable problems

Nonconvex continuous models for combinatorial optimization problems with application to satisfiability and node packing problems

We show how a large class of combinatorial optimization problems can be reformulated as a nonconvex minimization problem over the unit hyper cube with continuous variables. No additional constraints are required; all constraints are incorporated in the n onconvex objective function, which is a polynomial function. The application of the general transform to satisfiability and node packing problems is discussed, and various approximation algorithms are briefly reviewed. To give an indication of the strength of the proposed approaches, we conclude with some computational results on instances of the graph coloring problem

Relaxations of the satisfiability problem using semidefinite programming

We derive a semidefinite relaxation of the satisfiability (SAT) problem and discuss its strength. We give both the primal and dual formulation of the relaxation. The primal formulation is an eigenvalue optimization problem, while the dual formulation is a semidefinite feasibility problem. It is shown that using the relaxation, the notorious pigeon hole and mutilated chessboard problems are solved in polynomial time. As a byproduct we find a new `sandwich' theorem that is similar to Lov'asz' famous $vartheta$-function. Furthermore, using the semidefinite relaxation 2SAT problems are solved. By adding an objective function to the dual formulation, a specific class of polynomially solvable 3SAT instances can be identified. We conclude with discussing how the relaxation can be used to solve more general SAT problems and some empirical observations

Recognition of tractable satisfiability problems through balanced polynomial representations

AbstractWe consider a specific class of satisfiability (SAT) problems, the conjunctions of (nested) equivalencies (CoE). It is well known that CNF (conjunctive normal form) translations of CoE formulas are hard for branching and resolution algorithms. Tseitin proved that regular resolution requires a running time exponential in the size of the input. We review a polynomial time algorithm for solving CoE formulas, and address the problem of recognizing a CoE formula by its CNF representation. Making use of elliptic approximations of 3SAT problems, the so-called doubly balanced 3SAT formulas can be seen to be equivalent to CoE formulas. Subsequently, the notion of doubly balancedness is generalized by using polynomial representations of satisfiability problems, to obtain a general characterization of CoE formulas. We briefly address the problem of finding CoE subformulas, and finally the application of the developed theory to several DIMACS benchmarks is discussed