Joost P. Warners

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.

A two-phase algorithm for solving a class of hard satisfiability problemsfn1fn1Supported by the Dutch Organization for Scientific Research (NWO) under grant SION 612-33-001.

The DIMACS suite of satisfiability (SAT) benchmarks contains a set of instances that are very hard for existing algorithms. These instances arise from learning the parity function on 32bits. In this paper we develop a two-phase algorithm that is capable of solving these instances. In the first phase, a polynomially solvable subproblem is identified and solved. Using the solution to this problem, we can considerably restrict the size of the search space in the second phase of the algorithm, which is an extension of the well-known Davis-Putnam-Logemann-Loveland algorithm. We conclude with reporting on our computational results on the parity instances.

A potential reduction approach to the frequency assignment problem

The frequency assignment problem is the problem of assigning frequencies to transmission links such that either no interference occurs, or the amount of interference is minimized. We present an approximation algorithm for this problem that is inspired by Karmarkars interior point potential reduction approach to combinatorial optimization problems. A non convex quadratic model of the problem is developed, that is very compact as all interference constraints are incorporated in the objective function. Moreover, optimizing this model may result in finding multiple solutions to the problem simultaneouly. Several preprocessing techniques are discussed. We report on computational experience with both real-life and randomly generated instances.

Solving satisfiability problems using elliptic approximations - effective branching rules

Abstract An elliptic approximation of 3SAT problems is derived. It is used to derive branching rules for application in a Davis–Putnam–Logemann–Loveland branching & backtracking algorithm. Using the ellipsoid several well-known branching rules are rediscovered, but now they are obtained with a geometrical motivation. In fact, these rules can be considered to be approximations of the new rules we obtain, that make full use of the elliptic structure. These rules are more effective than the ‘old’ branching rules in terms of node counts. Extensive computational results are provided.

Potential reduction algorithms for structured combinatorial optimization problems

Recently Karmarkar proposed a potential reduction algorithm for binary feasibility problems. In this paper, a modified potential function that has more attractive properties is introduced. Furthermore, as the main result, for a specific class of binary feasibility problems a concise reformulation as nonconvex quadratic optimization problems is developed. We introduce a potential function to optimize the new model and report on computational experience with the graph coloring problem, comparing the performance of the three potential functions.

Bounds and fast approximation algorithms for binary quadratic optimization problems with application to MAX 2SAT

We consider binary convex quadratic optimization problems, particularly those arising from reformulations of well-known combinatorial optimization problems such as MAX 2SAT (and MAX CUT). A bounding and approximation technique is developed. This technique subsumes the spherical relaxation, while it can also be considered as a restricted variant of the semidefinite relaxation. Its complexity however is comparable to that of the first. It is shown how the quality of the obtained approximate solution can be measured. We conclude with extensive computational results on the MAX 2SAT problem, which show that good-quality solutions are obtained.

Solving Satisfiability Problems Using Elliptic Approximations. A Note on Volumes and Weights

In this note we propose to use the volume of elliptic approximations of satisfiability problems as a measure for computing weighting coefficients of clauses of different lengths. For random 3-SAT formula it is confirmed experimentally that, when applied in a DPLL algorithm with a branching strategy that is based on the ellipsoids as well, the weight deduced yields better results than the weights that are used in previous studies.