Using Branch-and-Bound with Constraint Satisfaction in Optimization Problems

Optimization Problems Stephen Beale Computing Research Laboratory Box 30001 New Mexico State University Las Cruces, New Mexico 88003 [email protected] Abstract This work1 integrates three related AI search techniques { constraint satisfaction, branch-and-bound and solution synthesis { and applies the result to constraint satisfaction problems for which optimal answers are required. This method has already been shown to work well in natural language semantic analysis (Beale, et al, 1996); here we extend the domain to optimizing graph coloring problems, which are abstractions of many common scheduling problems of interest. We demonstrate that the methods used here allow us to determine optimal answers to many types of problems without resorting to heuristic search, and, furthermore, can be combined with heuristic search methods for problems with excessive complexity. Introduction Optimization problems can be solved using a number of di erent techniques within the constraint satisfaction paradigm. Full lookahead (Haralick & Elliott, 1980) is popular because it is easy to implement and works on a large variety of problems. Tsang and Foster (1990) demonstrated that the Essex algorithms, a variation on Freuder's (1978) solution synthesis techniques, signi cantly outperform lookahead algorithms on the NQueens problem. Heuristic methods such as (Minton, et al, 1990) hold great promise for nding solutions to CSPs, but their heuristic nature preclude them from being used when optimal solutions are required. Our work follows along the line of Tsang and Foster and, earlier, Freuder, but instead of using constraints as the primary vehicle for reducing complexity at each stage of synthesis, we use branch-and-bound methods focused on the optimization aspect of the problem. Furthermore, we extend Tsang and Foster's use of variable ordering techniques such as \minimal bandwidth ordering" (MBO) to a more general partitioning of the input graph into fairly independent sub-problems. We demonstrate the utility of these techniques by solving 1Copyright c 1997, American Association for Arti cial Intelligence (www.aaai.org). All rights reserved. Research reported in this paper was supported in part by Contract MDA904-92-C-5189 from the U.S. Department of Defense. A B D C {r,y,g,b}