Cutting planes in constraint programming: An hybrid approach

In recent years, a growing number of attempts have been performed in order to integrate well known Operations Research(OR) techniques in Constraint Programming (CP) tools. The aim of the integration is to maintain the modelling facilities of the CP paradigm, while improving its performances by exploiting effective OR techniques. In our previous work, we proposed the use of optimization constraints [9], embedding a linear relaxation of the constraint itself and performing pruning on the basis of costs. In particular, domain values can be removed whenever it can be shown that their assignment will necessarily lead to solutions worse than the best solution found. In this setting, the use of cutting planes in global constraints allows to tighten the relaxation so as to infer more accurate bounds on the problem. We propose different ways of using cutting-planes in optimization constraints achieving different levels of tightness of the integration and pruning power. Even if the proposed technique is general, we use as testing application the Travelling Salesman Problem (TSPs) and its time constrained variant. Computational results compare different relaxations in terms of pruning achieved and computational complexity.