Branch-and-Cut Algorithms for Independent Set Problems: Integrality Gap and An Application to Protein Structure Alignment

We discuss the effectiveness of branch and cut for solving large instances of the independent set problem. Typical LP formulations, even strengthened by clique inequalities, yield poor bounds for this problem. We prove that a strong bound is obtained by the use of the so called “rank inequalities”, which generalize the clique inequalities. For some problems the clique inequalities imply the rank inequalities, and then a strong bound is guaranteed already by the simpler formulation. This is the case of the contact map overlap problem, which was proposed as a measure for protein structure alignments. We formalize this problem as a particular, large independent set problem which we solve