Detecting and Exploiting Permutation Structures in MIPs

Abstract

Many combinatorial optimization problems can be formulated as the search for the best possible permutation of a given set of objects, according to a given objective function. The corresponding MIP formulation is thus typically made of an assignment substructure, plus additional constraints and variables (as needed) to express the objective function. Unfortunately, the permutation structure is generally lost when the model is flattened as a mixed integer program, and state-of-the-art MIP solvers do not take full advantage of it. In the present paper we propose a heuristic procedure to detect permutation problems from their MIP formulation, and show how we can take advantage of this knowledge to speed up the solution process. Computational results on quadratic assignment and single machine scheduling problems show that the technique, when embedded in a state-of-the-art MIP solver, can indeed improve performance.