Abstract. In this paper we introduce MINIMAXSAT, a new Max-SAT solver that incorporates the best SAT and Max-SAT techniques. It can handle hard clauses (clauses of mandatory satisfaction as in SAT), soft clauses (clauses whose falsification is penalized by a cost as in Max-SAT) as well as pseudo-boolean objective functions and constraints. Its main features are: learning and backjumping on hard clauses; resolution-based and subtraction-based lower bounding; and lazy propagation with the two-watched literals scheme. Our empirical evaluation on a wide set of optimization benchmarks indicates that its performance is usually close to the best specialized alternative and, in some cases, even better. 1

In this paper we introduce MINIMAXSAT, a new Max-SAT solver that is built on top of MIN-ISAT+. It incorporates the best current SAT and Max-SAT techniques. It can handle hard clauses (clauses of mandatory satisfaction as in SAT), soft clauses (clauses whose falsification is penalized by a cost as in Max-SAT) as well as pseudo-boolean objective functions and constraints. Its main features are: learning and backjumping on hard clauses; resolution-based and substractionbased lower bounding; and lazy propagation with the two-watched literal scheme. Our empirical evaluation comparing a wide set of solving alternatives on a broad set of optimization benchmarks indicates that the performance of MINIMAXSAT is usually close to the best specialized alternative and, in some cases, even better. 1.

Given a Boolean satisfiability (Sat) problem whose variables have non-negative weights, the minimum-cost satisfiability (MinCostSat) problem finds a satisfying truth assignment that minimizes a weighted sum of the truth values of the variables. Many NP-optimization problems are either special cases of MinCostSat or can be transformed into MinCostSat efficiently. However, in the past, these problems have been largely considered in isolation. In this dissertation, we (1) classify existing Min-CostSat problems, (2) study factors affecting the performance of MinCostSat solvers, (3) propose algorithms for MinCostSat problems, and (4) implement and validate the performance of state-of-the-art solvers for special cases of MinCostSat, including set and binate covering, Max-Sat, and group-partial Max-Sat. We categorize MinCostSat problems as either native or non-native. Non-native problems can only be transformed into MinCostSat by adding slack variables. These problems include the Max-Sat, partial Max-Sat, and group-partial Max-Sat problems which have applications ranging from course assignment to FPGA detailed routing. Native problems are various sub-cases of MinCostSat. We further divide these into two

Integer and combinatorial optimization problems constitute a major challenge for algorithmics. They arise when a large number of discrete organizational decisions have to be made, subject to constraints and optimization criteria. This thesis

. For problems SAT and MAX SAT, local search algorithms are widely acknowledged as one of the most eective approaches. Most of the local search algorithms are based on the 1-ip neighborhood, which is the set of solutions obtainable by ipping the truth assignment of one variable. In this paper, we consider r-ip neighborhoods for r 2, and propose, for r = 2; 3, new implementations that reduce the number of candidates in the neighborhood without sacricing the solution quality. For 2-ip (resp., 3-ip) neighborhood, we show that its expected size is O(n + m) (resp., O(m + t 2 n)), which is usually much smaller than the original size O(n 2 ) (resp., O(n 3 )), where n is the number of variables, m is the number of clauses and t is the maximum number of appearances of one variable. Computational results tell that these estimates by the expectation well represent the real performance. These neighborhoods are then used under the framework of tabu search etc., and compa...

One of the challenges for the constraint satisfaction community has been to develop an automated approach to solving Constraint Satisfaction Problems (CSPs) rather than creating specific algorithms for specific problems. Much of this work has concentrated on the development and improvement of general purpose backtracking techniques. However, the success of relatively simple local search techniques on larger satisfiability problems [Selman et al. 1992] and CSPs such as the n-queens [Minton et al. 1992] has caused interest in applying local search to constraint satisfaction. In this thesis we look at the usefulness of constraint weighting as a local search technique for constraint satisfaction. The work is based on the clause weighting ideas of Selman and Kautz [1993] and Morris [1993] and applies, evaluates and extends these ideas from the satisfiability domain to the more general domain of CSPs. Specifically, the contributions of the thesis are: The introduction of a local search taxonomy. We examine the various better known local search techniques and recognise four basic strategies: restart, randomness, memory and weighting.

Abstract. Many real-world constraint satisfaction problems (CSPs) can be over-constrained but contain a set of mandatory or hard constraints that have to be satisfied for a solution to be acceptable. Recent research has shown that constraint weighting local search algorithms can be very effective in solving a variety of CSPs. However, little work has been done in applying such algorithms to over-constrained problems with hard constraints. The difficulty has been finding a weighting scheme that can weight unsatisfied constraints and still maintain the distinction between the mandatory and non-mandatory constraints. This paper presents a new weighting strategy that simulates the transformation of an over-constrained problem with mandatory constraints into an equivalent problem where all constraints have equal importance, but the hard constraints have been repeated. In addition, two dynamic constraint weighting schemes are introduced that alter the number of simulated hard constraint repetitions according to feedback received during the search. The dynamic constraint weighting algorithms are compared with two algorithms that maintain a fixed number of hard constraint repetitions, using a test bed of over-constrained timetabling and nurse rostering problems. The results show the dynamic strategies outperform both fixed repetition approaches. 1

For problems SAT and MAX SAT, local search algorithms are widely acknowledged as one of the most eective approaches. Most of the local search algorithms are based on the 1-ip neighborhood, which is the set of solutions obtainable by ipping the truth assignment of one variable. In this paper, we consider r-ip neighborhoods for r 2, and propose, for r = 2; 3, new implementations that reduce the number of candidates in the neighborhood without sacri cing the solution quality. For 2-ip (resp., 3-ip) neighborhood, we show that its expected size is O(n + m) (resp., O(m+ t n)), which is usually much smaller than the original size O(n ) (resp., O(n )), where n is the number of variables, m is the number of clauses and t is the maximum number of appearances of one variable. Computational results tell that these estimates by the expectation well represent the real performance.

In this note we present a formulation of two related combinatorial embedding problems concerning level graphs in terms of CNF -formulas. The rst problem is known as level planar embedding and the second as crossing-minimization-problem.

. The purpose of this paper is to speed up the local search algorithm for the CNF Satisfiability problem. Our basic strategy is to run some 10 5 independent search paths simultaneously using PVM on a vector supercomputer VPP800, which consists of 40 vector processors. Using the above parallelization and vectorization together with some improvement of data structure, we obtained 600-times speedup in terms of the number of flips the local search can make per second compared to the original GSAT by Selman and Kautz. We run our parallel GSAT for benchmark instances and compared the running time with those of existing SAT programs. We could observe an apparent benefit of parallelization: Especially, we were able to solve two instances that have never been solved before this paper. We also tested parallel local search for the SAT encoding of the class scheduling problem. Again we were able to get almost the best answer in reasonable time. 1 Introduction Local search is proba...