Edge Elimination in TSP Instances

TL;DR

This paper introduces graph theoretic results and an algorithm to eliminate edges that cannot be in any optimal TSP tour, significantly speeding up the solution process when combined with existing solvers.

Contribution

It presents a novel combinatorial algorithm for edge elimination in TSP instances, improving solution efficiency when integrated with Concorde.

Findings

Able to prove certain edges cannot be in optimal tours

Algorithm runs in O(n^2 log n) time for n-vertex instances

Solves a large TSPLIB instance over 11 times faster with Concorde

Abstract

The Traveling Salesman Problem is one of the best studied NP-hard problems in combinatorial optimization. Powerful methods have been developed over the last 60 years to find optimum solutions to large TSP instances. The largest TSP instance so far that has been solved optimally has 85,900 vertices. Its solution required more than 136 years of total CPU time using the branch-and-cut based Concorde TSP code [1]. In this paper we present graph theoretic results that allow to prove that some edges of a TSP instance cannot occur in any optimum TSP tour. Based on these results we propose a combinatorial algorithm to identify such edges. The runtime of the main part of our algorithm is $O(n^2 \log n)$ for an n-vertex TSP instance. By combining our approach with the Concorde TSP solver we are able to solve a large TSPLIB instance more than 11 times faster than Concorde alone.