Local Branching in a Constraint Programming Framework

Local branching is a general purpose heuristic method which searches locally around the best known solution by employing tree search. It has been successfully used in Mixed Integer Programming (MIP) where local branching constraints are used to model the neighborhood of an incumbent solution and improve the bound. The neighborhoods are obtained by linear inequalities in the MIP model so that MIP searches for the optimal solution within the Hamming distance of the incumbent solution. The linear constraints representing the neighborhood of incumbent solutions are called local branching constraints and are involved in the computation of the problem bound. Local branching is a general framework to effectively explore solution subspaces, making use of the state-of-the-art MIP solvers. The local branching framework is not specific to MIP: it can be integrated in any tree search strategy. In our research work we propose the integration of the local branching framework in Constraint Programming (CP). The local branching technique, as introduced in [Fis03], is a complete search method designed for providing solutions of better and better quality in the early stages of search by systematically defining and exploring large neighborhoods. On the other hand, the idea has been used mainly in an incomplete manner since [Fis03]: linear constraints defining large neighborhoods are iteratively added and the neighborhoods are explored, generally in a non-exhaustive way. When this is done within a local search method, the overall algorithm follows the spirit of both large neighborhood search [Sha98] and variable neighborhood search [Mla97]. The main peculiarity of local branching is that the neighborhoods and their exploration are general purpose. Integration of local search and CP aided tree search has been long advocated in the literature. See for instance Chapter 9 in [Mil03] and more recently the use of propagation within a large neighborhood search algorithm [Per04], the use of local search to speed up complete search [Sel06] and Beck’s solution-guided multi-point constructive search [Bec07]. The technique which resembles most to our work is the latter which makes use of the existing solutions to guide the search. We argue that integrating local branching in CP merges the advantages of the intensification and diversification mechanisms specific to local search methods, with constraint propagation that speeds up the neighborhood exploration by removing infeasible variable value assignments.