Separating a Superclass of Comb Inequalities in Planar Graphs

Many classes of valid and facet-inducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2-matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedure which, given an LP relaxation vector x∗, 7nds one or more inequalities in the class which are violated by x∗, or proves that none exist. Such algorithms are at the core of the highly successful branch-and-cut algorithms for the STSP. However, whereas polynomial time exact separation algorithms are known for subtour elimination and 2-matching inequalities, the complexity of comb separation is unknown. A partial answer to the comb problem is provided in this paper. We de7ne a generalization of comb inequalities and show that the associated separation problem can be solved e:ciently when the subgraph induced by the edges with x∗ e ¿0 is planar. The separation algorithm runs in O(n3) time, where n is the number of vertices in the graph.