Small Instance Relaxations for the Traveling Salesman Problem

We explore Small Instance Relaxations in branch-and-cut for the TSP. For small TSP instances of up to 10 cities all facet-defining inequalities of the associated polytopes are known. To exploit this pool of inequalities, we shrink a given TSP support graph to a small graph with at most 10 vertices, search for a violated inequality in the pool, and eventually lift it to obtain a cutting-plane. A Small Instance Relaxation (SIR) is an LP relaxation strengthened by such cutting- planes. We mainly shrink k-way cuts (k ≤ 10) of weight kλ/2 to obtain promising small graphs (λ is the mincut weight). For the separation in the low-dimensional space we solve a series of QAP instances. Padberg-Rinaldi shrinking criteria, graph isomorphism detection and facet class selection are applied to avoid unnecessary QAP computations. Our computational results show the usefulness of SIRs for the TSP. We compare SIRs with local cuts by Applegate et al.