On the H-Free Extension Complexity of the TSP

TL;DR

This paper investigates the extension complexity of TSP polytopes when certain well-known inequalities are excluded, demonstrating that the complexity remains exponential, thus highlighting the inherent computational difficulty.

Contribution

It proves that removing blossom or simple comb inequalities from the TSP polytope does not reduce its exponential extension complexity, and introduces a new subclass of comb inequalities for analysis.

Findings

Extension complexity remains exponential without blossom or simple comb inequalities.

Polytope formed by all comb inequalities also has exponential extension complexity.

Introduces (h,t)-uniform inequalities as a new analytical tool.

Abstract

It is known that the extension complexity of the TSP polytope for the complete graph $K_n$ is exponential in $n$ even if the subtour inequalities are excluded. In this article we study the polytopes formed by removing other subsets $\mathcal{H}$ of facet-defining inequalities of the TSP polytope. In particular, we consider the case when $\mathcal{H}$ is either the set of blossom inequalities or the 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 polytope formed by all comb inequalities is exponential. For our proofs, we introduce a subclass of comb inequalities, called $(h,t)$-uniform inequalities, which may be of independent interest.