Comb inequalities for typical Euclidean TSP instances

TL;DR

This paper demonstrates that augmenting the Held-Karp LP relaxation with all bounded comb inequalities does not significantly improve the approximation for Euclidean TSP, leading to exponential-time algorithms even on average cases.

Contribution

It proves that adding all bounded comb inequalities to the Held-Karp LP relaxation does not close the integrality gap in average cases, implying exponential complexity for related algorithms.

Findings

Integrality gap remains close to 1 with bounded comb inequalities.

Large classes of branch-and-cut algorithms are exponential-time for Euclidean TSP.

Even on random inputs, the LP relaxation with comb inequalities is not sufficient.

Abstract

We prove that even in average case, the Euclidean Traveling Salesman Problem exhibits an integrality gap of $(1+\epsilon)$ for $\epsilon>0$ when the Held-Karp Linear Programming relaxation is augmented by all comb inequalities of bounded size. This implies that large classes of branch-and-cut algorithms take exponential time for the Euclidean TSP, even on random inputs.