On the H-Free Extension Complexity of the TSP

It is known that the extension complexity of the TSP polytope for the complete graph Kn is exponential in n even if the subtour inequalities are excluded. In this article we study the polytopes formed by removing other subsets H of facet-defining inequalities of the TSP polytope. In particular, we consider the case when H is either the set of 2-matching inequalities or the set of simple comb inequalities. These inequalities are routinely used in cutting plane algorithms for the TSP. We show that the extension complexity remains exponential even if we exclude these inequalities. In addition we show that the extension complexity of the polytope formed by all comb inequalities is exponential. For our proofs, we introduce a subclass of comb inequalities, called (h, t)-uniform comb inequalities, which may be of independent interest in the study of the TSP.