Shorter Tours by Nicer Ears: 7/5-approximation for graphic TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs

TL;DR

This paper presents improved polynomial-time approximation algorithms for the graphic TSP, connected-$T$-join, and 2-edge-connected subgraph problems, achieving ratios of 7/5, 3/2, and 4/3 respectively, using novel ear-decomposition techniques.

Contribution

It introduces new approximation algorithms with better guarantees for graphic TSP and related problems, employing innovative ear-decomposition methods.

Findings

Graphic TSP approximation ratio improved to 7/5

Connected-$T$-join problem approximated with ratio 3/2

2-edge-connected subgraph approximation ratio improved to 4/3

Abstract

We prove new results for approximating the graphic TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graphic TSP itself, we improve the approximation ratio to 7/5. For a generalization, the connected-$T$-join problem, we obtain the first nontrivial approximation algorithm, with ratio 3/2. This contains the graphic $s$-$t$-path-TSP as a special case. Our improved approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4/3. The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios.