For the last ten years, a significant amount of work in the constraint community has been devoted to the improvement of complete methods for solving soft constraints networks. We wanted to see how recent progress in the weighted CSP (WCSP) field could compete with other approaches in related fields.

Artificial Intelligence, to appear 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 Boolean formula. A branch and bound algorithm based on the Davis-Putnam-Logemann-Loveland procedure (DPLL) is one of the most competitive exact algorithms for solving max-SAT. In this paper, we propose and investigate a number of strategies for max-SAT. The first strategy is a set of unit propagation or unit resolution rules for max-SAT. We summarize three existing unit propagation rules and propose a new one based on a nonlinear programming formulation of max-SAT. The second strategy is an effective lower bound based on linear programming (LP). We show that the LP lower bound can be made effective as the number of clauses increases. The third strategy consists of a a binary-clause first rule and a dynamicweighting variable ordering rule, which are motivated by a thorough analysis of two existing well-known variable orderings. Based on the analysis of these strategies, we develop an exact solver for both max-SAT and weighted max-SAT. Our experimental results on random problem instances and many instances from the max-SAT libraries show that our new solver outperforms most of the existing exact max-SAT solvers, with orders of magnitude of improvement in many cases.

Is it possible to predict how long an algorithm will take to solve a previously-unseen instance of an NP-complete problem? If so, what uses can be found for models that make such predictions? This paper provides answers to these questions and evaluates the answers experimentally. We propose the use of supervised machine learning to build models that predict an algorithm’s runtime given a problem instance. We discuss the construction of these models and describe techniques for interpreting them to gain understanding of the characteristics that cause instances to be hard or easy. We also present two applications of our models: building algorithm portfolios that outperform their constituent algorithms, and generating test distributions that emphasize hard problems. We demonstrate the effectiveness of our techniques in a case study of the combinatorial auction winner determination problem. Our experimental results show that we can build very accurate models of an algorithm’s running time, interpret our models, build an algorithm portfolio that strongly outperforms the best single algorithm, and tune a standard benchmark suite to generate much harder problem instances.

Backbone variables are the elements that are common to all optimal solutions of a problem instance. We call variables that are absent from every optimal solution fat variables.

In recent years, there has been much interest in phase transitions of combinatorial problems.

received the Schupf fellowship, which generously supports PhD students with leadership potential. In addition to the sponsors of these grants, I am grateful to all of the administrators who helped insure that my funding was transferred in a timely fashion. I would also like to acknowledge all of my collaborators. First, the Maverick and DAI labs at Bar-Ilan were always my academic home. All of the lab members are talented researchers who stimulated many fruitful discussions. I would especially like to mention Noa Segel-Argamon, Michal Chalamish, Ariel Felner, Meirav Hadad, Meir Kalech, Efrat Manister, Dudi Sarne, Osher Yadgar and Aner Yarden for their help throughout my time at Bar-Ilan. I would like to single out Noa who was extremely helpful in formulating several concepts in Chapters 2 and 3. The NSF and DARPA projects I worked on introduced me to many interesting people outside of Bar-Ilan. I am indebted to Barbara Grosz of Harvard who spent many hours giving encouragement, and stimulating many interesting conversations. Willem-Jan van Hoeve of Cornell developed the scheduler used in Chapter 4 of this

This chapter covers research in constraint programming (CP) and related areas involving random problems. Such research has played a significant role in the development of more efficient and effective algorithms, as well as in understanding the source of hardness in solving combinatorially challenging problems. Random problems have proved useful in a number of different ways. Firstly, they provide a relatively “unbiased ” sample for benchmarking algorithms. In the early days of CP, many algorithms were compared using only a limited sample of problem instances. In some cases, this may have lead to premature conclusions. Random problems, by comparison, permit algorithms to be tested on statistically significant samples of hard problems. However, as we outline in the rest of this chapter, there remain pitfalls waiting the unwary in their use. For example, random problems may not contain structures found in many real world problems, and these structures can make problems much easier or much harder to solve. As a second example, the process of generating random problems may itself be “flawed”, giving problem instances which are not, at least asymptotically, combinatorially hard. Random problems have also provided insight into problem hardness. For example, the influential paper by Cheeseman, Kanefsky and Taylor [12] highlighted the computational difficulty of problems which are on the “knife-edge ” between satisfiability and unsatisfiability [84]. There is even hope within certain quarters that random problems may be one of the links in resolving the P=NP question. Finally, insight into problem hardness provided by random problems has helped inform the design of better algorithms and heuristics. For example, the design of a number of branching heuristics for the Davis Logemann Loveland satisfiability (DPLL) procedure has been heavily influenced by the hardness of random problems. As a second example, the rapid randomization and restart (RRR) strategy [45, 44] was motivated by the discovery of heavy-tailed runtime distributions in backtracking style search procedures on random quasigroup completion problems.

We study the backbone of the travelling salesperson optimization problem. We prove that it is intractable to approximate the backbone with any performance guarantee, assuming that P�=NP and there is a limit on the number of edges falsely returned. Nevertheless, in practice, it appears that much of the backbone is present in close to optimal solutions. We can therefore often find much of the backbone using approximation methods based on good heuristics. We demonstrate that such backbone information can be used to guide the search for an optimal solution. However, the variance in runtimes when using a backbone guided heuristic is large. This suggests that we may need to combine such heuristics with randomization and restarts. In addition, though backbone guided heuristics are useful for finding optimal solutions, they are less help in proving optimality. 1

Multi-agent systems are particularly appropriate for resource allocation, but configuring them for efficient operation requires understanding their dynamics. Concepts from statistical physics, such as phase transitions, can help. In decision problems such as constraint satisfaction, such transitions exhibit an easy-hard-easy effort profile, so that highly overconstrained problems are easier to solve than those near the transition. The conventional wisdom is that the profile in optimization problems such as resource allocation is monotonic, becoming more difficult as constraints increase. Contrary to this lore, we exh ibit an easy-hard-easy profile in a multi-agent resource allocation problem. We compare Autonomous Negotiating Teams (ANT) systems that exhibit such a profile with others that do not and offer insights as to when such behavior can be expected and why it is desirable from a practical perspective.