Reassembling Trees for the Traveling Salesman

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 Sebo's previously best approximation ratio of $\frac{8}{5}$.