An Experimental Evaluation of the Best-of-Many Christofides' Algorithm for the Traveling Salesman Problem

TL;DR

This paper empirically evaluates the Best-of-Many Christofides' algorithm for TSP, demonstrating its superior performance over the classic Christofides' algorithm through various sampling schemes.

Contribution

It provides an experimental comparison of different sampling methods for the Best-of-Many Christofides' algorithm, highlighting the effectiveness of maximum entropy sampling.

Findings

All tested methods outperform Christofides' algorithm.

Maximum entropy sampling yields particularly strong results.

Other sampling schemes also show significant improvements.

Abstract

Recent papers on approximation algorithms for the traveling salesman problem (TSP) have given a new variant on the well-known Christofides' algorithm for the TSP, called the Best-of-Many Christofides' algorithm. The algorithm involves sampling a spanning tree from the solution the standard LP relaxation of the TSP, subject to the condition that each edge is sampled with probability at most its value in the LP relaxation. One then runs Christofides' algorithm on the tree by computing a minimum-cost matching on the odd-degree vertices in the tree, and shortcutting the resulting Eulerian graph to a tour. In this paper we perform an experimental evaluation of the Best-of-Many Christofides' algorithm to see if there are empirical reasons to believe its performance is better than that of Christofides' algorithm. Furthermore, several different sampling schemes have been proposed; we implement several different schemes to determine which ones might be the most promising for obtaining improved performance guarantees over that of Christofides' algorithm. In our experiments, all of the implemented methods perform significantly better than the Christofides' algorithm; an algorithm that samples from a maximum entropy distribution over spanning trees seems to be particularly good, though there are others that perform almost as well.