Richard J. Wallace

Partial constraint satisfaction

Abstract A constraint satisfaction problem involves finding values for variables subject to constraints on which combinations of values are allowed. In some cases it may be impossible or impractical to solve these problems completely. We may seek to partially solve the problem, in particular by satisfying a maximal number of constraints. Standard backtracking and local consistency techniques for solving constraint satisfaction problems can be adapted to cope with, and take advantage of, the differences between partial and complete constraint satisfaction. Extensive experimentation on maximal satisfaction problems illuminates the relative and absolute effectiveness of these methods. A general model of partial constraint satisfaction is proposed.

Stable Solutions for Dynamic Constraint Satisfaction Problems

An important extension of constraint technology involves problems that undergo changes that may invalidate the current solution. Previous work on dynamic problems sought methods for efficiently finding new solutions. We take a more proactive approach, exploring methods for finding solutions more likely to remain valid after changes that temporarily alter the set of valid assignments (stable solutions). To this end, we examine strategies for tracking changes in a problem and incorporating this information to guide search to solutions that are more likely to be stable. In this work search is carried out with a min-conflicts hill climbing procedure, and information about change is used to bias value selection, either by distorting the objective function or by imposing further criteria on selection. We study methods that track either value losses or constraint additions, and incorporate information about relative frequency of change into search. Our experiments show that these methods are generally effective in finding stable solutions, and in some cases handle the tradeoff between solution stability and search efficiency quite well. In addition, we identify one condition in which these methods markedly reduce the effort to find a stable solution.

Directed Arc Consistency Preprocessing

A constraint satisfaction problem (CSP) may be overconstrained and not admit a complete solution. Optimal solutions to such partial constraint satisfaction problems (PCSPs), in which a maximum number of constraints are satisfied, can be found using branch and bound variants of CSP algorithms. Earlier work has shown how information gained through local consistency checking during preprocessing can be used to enhance search through value ordering heuristics and local lower bound calculations that involve only neighboring variables. The present work describes a family of strategies based on directed arc consistency testing during preprocessing. With this approach inconsistency counts associated with each value (the number of domains that offer no support for that value) are obtained that are non-redundant, since they are unidirectional. They can, therefore, be used to obtain global lower bounds that involve the entire set of variables. By computing directed arc consistency in each direction, full arc- inconsistency counts can also be obtained, thus retaining the benefits of full arc consistency checking, while improving lower bound calculations. Retrospective and prospective algorithms that incorporate the results of directed arc consistency checking are described. Tests with random problems show improvements, sometimes marked, over the best branch and bound PCSP algorithms heretofore described.

Suggestion Strategies for Constraint-Based Matchmaker Agents

In this paper we describe a paradigm for content-focused matchmaking, based on a recently proposed model for constraint acquisition and satisfaction. Matchmaking agents are conceived as constraint-based solvers that interact with other, possibly human, agents (Customers). The Matchmaker provides potential solutions (suggestions) based on partial knowledge, while gaining further information about the problem itself from the other agent through the latters evaluation of these suggestions. The dialog between Matchmaker and Customer results in iterative improvement of solution quality, as demonstrated in simple simulations. We also show empirically that this paradigm supports suggestion strategies for finding acceptable solutions more efficiently or for increasing the amount of information obtained from the Customer. This work also indicates some ways in which the tradeoff between these two metrics for evaluating performance can be handled.

Heuristic Methods for Over-Constrained Constraint Satisfaction Problems

Heuristic repair methods have successfully solved constraint satisfaction problems (CSPs) and satisfiability problems (SAT) that are too large to be solved by complete algorithms. In this paper we develop methods for testing the efficiency and quality of solution returned by these methods when applied to overconstrained CSPs and SAT. The key strategy is to test heuristic methods on problems of moderate size with known optimal distances (number of constraint violations), as determined with complete algorithms. This allows us to determine whether heuristic methods find optimal distances and allows us to carry out more incisive analyses of efficiency when different strategies are incorporated into these methods and parameter values are varied. The present work tested the min-conflicts algorithm with CSPs, either alone or in combination with walk, reset or tabu strategies. SAT was tested with GSAT and walk-SAT. The best results for min-conflicts were found with the walk strategy, when the probability of random assignment was set at 0.10 or 0.15. Both GSAT and walk-SAT readily found optimal solutions for 3-SAT, the latter being somewhat faster overall.

Analysis of heuristic methods for partial constraint satisfaction problems

Problems that do not have complete solutions occur in many areas of application of constraint solving. Heuristic repair methods that have been used successfully on complete CSPs can also be used on overconstrained problems. A difficulty in analyzing their performance is the uncertainty about the goodness of solutions returned in relation to the optimal (best possible) solutions. This difficulty can be overcome by testing these procedures on problems that can be solved by complete methods, which return certifiably optimal solutions. With this experimental strategy, comparative analyses of hill-climbing methods were carried out using anytime curves that could be compared with known optima. In addition, extensive analysis of parameter values for key strategies such as random walk and restarting could be done precisely and efficiently by allowing local search to run until a solution was discovered that was known to be optimal, based on earlier tests with complete methods. An important finding is that a version of min-conflicts that incorporates the random walk strategy, with a good value for the walk probability appears to be as efficient in this domain as several of the more elaborate methods for improving local search that have been proposed in recent years. n nThis material is based on work supported by the National Science Foundation under Grant Nos. IRI-9207633. and IRI-9504316. Some of this material was presented at the Workshop on Overconstrained Systems at CP95.

Anytime algorithms for constraint satisfaction and SAT problems

The constraint satisfaction problem (CSP) is a potential area of application for anytime methods. In this work we derive anytime curves using a partial constraint satisfaction framework that encompasses problems with complete solutions and those that allow only partial solutions of varying quality. In either case, the curves should converge on optimal solutions with respect to some measure of cost (here, the number of violated constraints). Binary CSPs and k-satisfiability problems were tested, using heuristic repair and branch and bound methods. Curves for heuristic methods either start at a lower level than curves for branch and bound (min-conflicts with binary CSPs) or have a steeper initial descent (GSAT with k-SAT problems). Techniques for randomization such as random walks or restarting with a new random solution appear to be necessary with heuristic procedures for complete convergence to an optimal solution. Branch and bound algorithms are usefully employed in tandem with heuristic methods, especially to verify optimality and, therefore, the quality of solution returned by the latter.

Partial Constraint Satisfaction

A constraint satisfaction problem involves finding values for variables subject to constraints on which combinations of values are allowed. In some cases it may be impossible or impractical to solve these problems completely. We may seek to partially solve the problem, in particular by satisfying a maximal number of constraints. Standard backtracking and local consistency techniques for solving constraint satisfaction problems can be adapted to cope with, and take advantage of, the differences between partial and complete constraint satisfaction. Extensive experimentation on maximal satisfaction problems illuminates the relative and absolute effectiveness of these methods. A general model of partial constraint satisfaction is proposed.

Enhancing Maximum Satisfiablility Algorithms with Pure Literal Strategies

Maximum satisfiability (MAX-SAT) is an extension of satisfiability (SAT), in which a partial solution is sought that satisfies the maximum number of clauses in a logical formula. In recent years there has been an growing interest in this and other types of over-constrained problems. Branch and bound extensions of the Davis-Putnam algorithm can return guaranteed optimal solutions to these problems. Earlier work did not make use of a pure literal rule because it appealed to be inefficient here, as for traditional SAT. However, arguments can be adduced to show that pure literals are likely to appear during search for MAX-2SAT, so that fixation of their variables may be effective here. The present work confirms this and also shows that a value ordering heuristic involving literals that are monotone in unit open clauses can be very effective, operating somewhat independently of the ordinary fixation of fully monotone literals. Alone or together, these pure literal strategies can produce improvements of an order of magnitude or more when combined with versions of Davis-Putnam studied in earlier work, sometimes solving problems of considerable size.

Object retrieval by Norway rats as a framework for preference and choice.

Object retrieval by Norway rats consists of locomotion to an object, object seizure, carriage to a refuge and release. These studies examined choices in this sequence and object preferences (relative numbers retrieved) for objects varying in sweetness. In Experiment 1, rats could retrieve two types of object on each trial, drawn from a set of three, each placed in quantity in one arm of a Y-maze connected to the home cage. After a period of learning, a stable preference hierarchy was demonstrated that depended on choices in different parts of the retrieval sequence i.e. decision making was broadly distributed. For the middle object in the hierarchy, response probabilities varied with object pairing, suggesting that rats responded in terms of object-sets. Preferences also involved rapid updating at the beginning of each trial (working memory) manifested in shifts in responding in all segments that supported consistent choices. Experiments 2-3, with an empty arm as an extra alternative, indicated that visits to an arm with less-preferred objects represented exploration interspersed with retrieval. Preference and choice are fundamental to object retrieval in this species and choice consistency in a dynamic environment is based on learning and memory.