A (Slightly) Improved Bound on the Integrality Gap of the Subtour LP for TSP

TL;DR

This paper improves the upper bound on the integrality gap of the subtour LP for TSP, showing it is at most 1.5 minus a tiny epsilon, and provides a better randomized approximation for the 2-edge-connected multi-subgraph problem.

Contribution

It establishes a slightly improved bound on the integrality gap of the subtour LP for TSP and introduces a better randomized approximation algorithm for the related problem.

Findings

Expected cost of max entropy algorithm is at most (3/2 - epsilon) times optimal

Integrality gap of the subtour LP is at most (3/2 - epsilon)

Provides a randomized (3/2 - epsilon) approximation for the 2-edge-connected multi-subgraph problem

Abstract

We show that for some $\epsilon > 10^{-36}$ and any metric TSP instance, the max entropy algorithm returns a solution of expected cost at most $\frac{3}{2}-\epsilon$ times the cost of the optimal solution to the subtour elimination LP. This implies that the integrality gap of the subtour LP is at most $\frac{3}{2}-\epsilon$. This analysis also shows that there is a randomized $\frac{3}{2}-\epsilon$ approximation for the 2-edge-connected multi-subgraph problem, improving upon Christofides' algorithm.