Duality in Optimization and Constraint Satisfaction

We show that various duals that occur in optimization and constraint satisfaction can be classified as inference duals, relaxation duals, or both. We discuss linear programming, surrogate, Lagrangean, superadditive, and constraint duals, as well as duals defined by resolution and filtering algorithms. Inference duals give rise to nogood-based search methods and sensitivity analysis, while relaxation duals provide bounds. This analysis shows that duals may be more closely related than they appear, as are surrogate and Lagrangean duals. It also reveals common structure between solution methods, such as Benders decomposition and Davis-Putnam-Loveland methods with clause learning. It provides a framework for devising new duals and solution methods, such as generalizations of mini-bucket elimination. 1 Two Kinds of Duals Duality is perennial theme in optimization and constraint satisfaction. Wellknown optimization duals include the linear programming (LP), Lagrangean, surrogate, and superadditive duals. The constraint satisfaction literature discusses constraint duals as well as search methods that are closely related to duality. These many duals can be viewed as falling into two classes: inference duals and relaxation duals [12]. The two classes represent quite different concepts of duality. This is perhaps not obvious at first because the traditional optimization duals just mentioned can be interpreted as both inference and relaxation duals. Classifying duals as inference or relaxation duals reveals relationships that might not otherwise be noticed. For instance, the surrogate and Lagrangean duals do not seem closely related, but by viewing them as inference duals rather than relaxation duals, one sees that they are identical except for a slight alteration in the type of inference on which they are based. A general analysis of duality can also unify some existing solution methods and suggest new ones. Inference duals underlie a number of nogood-based search methods and techniques for sensitivity analysis. For instance, Benders decomposition and Davis-Putnam-Loveland methods with clause learning, which appear unrelated, are nogood-based search methods that result from two particular inference duals. Since any inference method defines an inference dual, one can in principle devise a great variety inference duals and investigate the nogoodbased search methods that result. For example, filtering algorithms can be seen as inference methods that define duals and give rise to new search methods, such as decomposition methods for planning and scheduling. Relaxation duals underlie a variety of solution methods that are based on bounding the objective function. A relaxation dual solves a class of problem relaxations that are parameterized by “dual variables,” in order to obtain a tight bound on the objective function value. The LP, surrogate, Lagrangean, and superadditive duals familiar to the optimization literature are relaxation duals. A constraint dual is not precisely a relaxation dual but immediately gives rise to one that generalizes mini-bucket elimination methods. Inference and relaxation duals are precise expressions of two general problemsolving strategies. Problems are often solved by a combination of search and inference; that is, by searching over values of variables, which can yield a certificate of feasibility for the original (“primal”) problem, and by simultaneously drawing inferences from constraints, which can yield a certificate of optimality by solving the dual problem. A problem belongs to NP when the primal solution has polynomial size and to co-NP when the dual solution has polynomial size. Problems can also be solved by a combination of search and relaxation; that is, by enumerating relaxations and solving each. The relaxation dual is one way of doing this, since it searches over values of dual variables and solves the relaxation corresponding to each value.