Reassembling trees for the traveling salesman

TL;DR

This paper introduces a reassembling technique for spanning trees in TSP approximation algorithms, leading to improved approximation ratios by exchanging edges before sampling trees.

Contribution

It proposes a novel edge-exchanging reassembling method for spanning trees that enhances approximation guarantees for the metric s-t-path TSP.

Findings

Improves the approximation ratio for s-t-path TSP beyond 8/5.

Provides a deterministic polynomial-time algorithm implementing the reassembling technique.

Demonstrates the effectiveness of reassembling trees in TSP approximation algorithms.

Abstract

Many recent approximation algorithms for different variants of the traveling salesman problem (asymmetric TSP, graph TSP, s-t-path TSP) exploit the well-known fact that a solution of the natural linear programming relaxation can be written as convex combination of spanning trees. The main argument then is that randomly sampling a tree from such a distribution and then completing the tree to a tour at minimum cost yields a better approximation guarantee than simply taking a minimum cost spanning tree (as in Christofides' algorithm). We argue that an additional step can help: reassembling the spanning trees before sampling. Exchanging two edges in a pair of spanning trees can improve their properties under certain conditions. We demonstrate the usefulness for the metric s-t-path TSP by devising a deterministic polynomial-time algorithm that improves on Seb\H{o}'s previously best approximation ratio of 8/5.