Approximating Graphic TSP by Matchings

TL;DR

This paper introduces a new matching-based framework for approximating the metric TSP, achieving improved approximation ratios and matching conjectured bounds for certain graph classes.

Contribution

The paper presents a novel approach using matchings for metric TSP approximation, including a 1.461-approximation for graphic TSP and matching the integrality gap for specific graph classes.

Findings

Achieves a 1.461-approximation for graphic TSP.

Shows the integrality gap matches 4/3 for certain graph classes.

Provides a 1.586-approximation for the traveling salesman path problem.

Abstract

We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges in order to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graph-TSP), the approach yields a 1.461-approximation algorithm with respect to the Held-Karp lower bound. For graph-TSP restricted to a class of graphs that contains degree three bounded and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio 4/3. The framework allows for generalizations in a natural way and also leads to a 1.586-approximation algorithm for the traveling salesman path problem on graphic metrics where the start and end vertices are prespecified.