Local elimination in the traveling salesman problem

TL;DR

This paper presents an implementation of a combinatorial elimination technique for the TSP, combined with LP reduction, significantly reducing graph sizes in large instances and aiding in solving previously unsolved problems.

Contribution

It introduces an efficient implementation of Hougardy and Schroeder's elimination technique, combined with LP reduction, to prune edges in large TSP instances.

Findings

Edge sets reduced to under 3n in most instances

Graph size decreased to under 2.5n in large unsolved instances

Effective in solving large-scale TSP problems

Abstract

Hougardy and Schroeder (WG 2014) proposed a combinatorial technique for pruning the search space in the traveling salesman problem, establishing that, for a given instance, certain edges cannot be present in any optimal tour. We describe an implementation of their technique, employing an exact TSP solver to locate k-opt moves in the elimination process. In our computational study, we combine LP reduced-cost elimination together with the new combinatorial algorithm. We report results on a set of geometric instances, with the number of points n ranging from 3,038 up to 115,475. The test set includes all TSPLIB instances having at least 3,000 points, together with 250 randomly generated instances, each with 10,000 points, and three currently unsolved instances having 100,000 or more points. In all but two of the test instances, the complete-graph edge sets were reduced to under 3n edges. For the three large unsolved instances, repeated runs of the elimination process reduced the graphs to under 2.5n edges.