Bounds-Consistent Local Search

Abstract

With this work we present a hybrid approach to solve large-scale constraint satisfaction and optimization problems. The method proposed can be termed Bounds-Consistent Local Search. It is inspired by the success of a randomized algorithm for the Boolean Satisfiability (SAT) problem called UnitWalk. We have adapted the algorithm UnitWalk to integer programs. The search space is pruned through propagation; particularly we use Bounds-Consistency (BC). In this way we combine Local Search which performs well on large combinatorial optimization problems, and consistency propagation methods used to solve constraint problems. Unit-Walk is a simple algorithm that performs well on different instances of hard SAT problems. It has given best results on industrial problems in SAT competition. It also has been proved it is Probabilistically Approximately Complete (PAC); it means that it succeeds with probability one without restarting for initial assignment. We opted to use bounds-consistency propagation for linear constraints for two reasons. Firstly, because bounds-consistency implies hyper-arc consistency on integer linear inequalities. Hyper-arc consistency is a strong form of consistency, and linear inequalities are an expressive form of constraint that has already been used to model many problems including Multiple Sequence Alignment problem (MSA) from Bioinformatics. Secondly, large domains do not need to be explicitly represented, which saves a lot of memory and reduces runtime overheads. With BC we need only maintain two numbers per integer variable: an upper and a lower bound. BC can be also applied to non-linear constraints such as all-different, which we plan to deal with in future work. The algorithm starts with a random assignment for the variables, and then explores the search space by randomly choosing the variable to be instantiated. It performs bounds propagation before and during search. If domain wipe-out occurs then it restarts the search, using previous successful assignments to guide the selection of domain values. We are improving the algorithm with new heuristics. We have developed a dynamic prioritization heuristic that uses the information gained during the search in order to set up variables’ selecting ordering. This prioritization is updated at each new search, and is inspired by an another algorithm called Squeaky Wheel Optimization.

This work has been supported by Science Foundation Ireland under Grant 04/BR/CS0355.