Lagrangian Decomposition via Sub-problem Search

One of the critical issues that affect the efficiency of branch and bound algorithms in Constraint Programming is how strong a bound on the objective function can be inferred at each search node. The stronger the bound that can be inferred, the earlier failed subtrees can be detected, leading to an exponentially smaller search tree. Normal CP solvers are only capable of inferring a bound on the objective function via propagating the problem constraints. Unfortunately, for many problem classes, this does not yield a very strong bound. Recently, Lagrangian decomposition methods have been adapted and applied to Constraint Programming in order to yield stronger bounds on the objective function. While these methods yield some success, they are somewhat limited in the types of problems they can be effectively applied to. In particular, the set of constraints has to be divided into subsets such that each subset can be solved efficiently via a specialized propagator, e.g., consists of a knapsack problem, or a cost-MDD problem. For many more practical problem classes, such a division of constraints is simply not possible and thus those methods cannot be applied. In this paper, we propose a Lagrangian decomposition method where the sub-problems are solved via search rather than through a specialized propagator. This has the benefit that the method can be applied to a much wider range of problems. We present experiments to show the effectiveness of our method.