Consistency for 0-1 Programming

Concepts of consistency have long played a key role in constraint programming but never developed in integer programming (IP). Consistency nonetheless plays a role in IP as well. For example, cutting planes can reduce backtracking by achieving various forms of consistency as well as by tightening the linear programming (LP) relaxation. We introduce a type of consistency that is particularly suited for 0-1 programming and develop the associated theory. We define a 0-1 constraint set as LP-consistent when any partial assignment that is consistent with its linear programming relaxation is consistent with the original 0-1 constraint set. We prove basic properties of LP-consistency, including its relationship with Chvatal-Gomory cuts and the integer hull. We show that a weak form of LP-consistency can reduce or eliminate backtracking in a way analogous to k-consistency but is easier to achieve. In so doing, we identify a class of valid inequalities that can be more effective than traditional cutting planes at cutting off infeasible 0-1 partial assignments

Mathematical programming embeddings of logic

"February 20th, 1998"--T.p. -- "June, 1998"--Cover.Includes bibliographical references (p. 21-23).Supported in part by the U.S. Army. DAAL03-92-G-0115 Supported in part by a Center for Intelligent Control Systems grant from Siemens AG.Vivek S. Borker ... [et al.

An Efficient Algorithm for Maximum Boolean Satisfiability Based on Unit Propagation, Linear Programming, and Dynamic Weighting

Maximum Boolean satisfiability (max-SAT) is the optimization counterpart of Boolean satisfiability (SAT), in which a variable assignment is sought to satisfy the maximum number of clauses in a logical formula. A branch-and-bound algorithm based on the Davis-Putnam-Logemann-Loveland procedure (DPLL) is one of the most efficient complete algorithms for solving max-SAT. In this paper, We propose and investigate a number of new strategies for max-SAT. Our first strategy is a set of unit propagation rules for max-SAT. As unit propaga-tion is a very efficient strategy for SAT, we show that it can be extended to max-SAT, and can greatly improve the performance of an extended DPLL-based algorithm. Our second strategy is an effective lookahead heuristic based on linear programming. We show that the LP heuristic can be made effective as the number of clauses increases. Our third strategy is a dynamic-weight variable ordering, which is based on a thorough analysis of two well-known existing branching rules. Based on the analysis of these strategies, we develop an integrated, constrainedness-sensitive max-SAT solver that is able to dynamically adjust strategies ac-cording to problem characteristics. Our experimental results on random max-SAT and some instances from the SATLIB show that our new solver outperforms most of the existing com-plete max-SAT solvers, with orders of magnitude of improvement in many cases

Analyzing Satisfiability and Refutability in Selected Constraint Systems

This dissertation is concerned with the satisfiability and refutability problems for several constraint systems. We examine both Boolean constraint systems, in which each variable is limited to the values true and false, and polyhedral constraint systems, in which each variable is limited to the set of real numbers R in the case of linear polyhedral systems or the set of integers Z in the case of integer polyhedral systems. An important aspect of our research is that we focus on providing certificates. That is, we provide satisfying assignments or easily checkable proofs of infeasibility depending on whether the instance is feasible or not. Providing easily checkable certificates has become a much sought after feature in algorithms, especially in light of spectacular failures in the implementations of some well-known algorithms. There exist a number of problems in the constraint-solving domain for which efficient algorithms have been proposed, but which lack a certifying counterpart. When examining Boolean constraint systems, we specifically look at systems of 2-CNF clauses and systems of Horn clauses. When examining polyhedral constraint systems, we specifically look at systems of difference constraints, systems of UTVPI constraints, and systems of Horn constraints. For each examined system, we determine several properties of general refutations and determine the complexity of finding restricted refutations. These restricted forms of refutation include read-once refutations, in which each constraint can be used at most once; literal-once refutations, in which for each literal at most one constraint containing that literal can be used; and unit refutations, in which each step of the refutation must use a constraint containing exactly one literal. The advantage of read-once refutations is that they are guaranteed to be short. Thus, while not every constraint system has a read-once refutation, the small size of the refutation guarantees easy checkability