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 theSymmetric Travelling Salesman Problem (STSP), includingsubtour elimination, 2-matching andcomb inequalities. For a given class of inequalities, anexact separation algorithm is a procedure which, given an LP relaxation vectorx*, finds one or more inequalities in the class which are violated byx*, or proves that none exist. Such algorithms are at the core of the highly successfulbranch-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.Apartial answer to the comb problem is provided in this paper. We define ageneralization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges withx*e>0 is planar. The separation algorithm runs in O( n3) time, wheren is the number of vertices in the graph.

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