Dancing to the State of the Art? How Candidate Lists Influence LKH for Solving the Traveling Salesperson Problem

TL;DR

This paper improves the LKH heuristic for the Traveling Salesperson Problem by integrating candidate list initialization strategies, significantly reducing timeouts and enhancing performance compared to the current state of the art.

Contribution

It introduces a novel candidate list initialization method based on Hamiltonian circuits and incorporates it into an enhanced restart version of LKH, leading to substantial efficiency gains.

Findings

Fewer timeouts with the new method

Order of magnitude faster performance

Better solution quality on challenging instances

Abstract

Solving the Traveling Salesperson Problem (TSP) remains a persistent challenge, despite its fundamental role in numerous generalized applications in modern contexts. Heuristic solvers address the demand for finding high-quality solutions efficiently. Among these solvers, the Lin-Kernighan-Helsgaun (LKH) heuristic stands out, as it complements the performance of genetic algorithms across a diverse range of problem instances. However, frequent timeouts on challenging instances hinder the practical applicability of the solver. Within this work, we investigate a previously overlooked factor contributing to many timeouts: The use of a fixed candidate set based on a tree structure. Our investigations reveal that candidate sets based on Hamiltonian circuits contain more optimal edges. We thus propose to integrate this promising initialization strategy, in the form of POPMUSIC, within an efficient restart version of LKH. As confirmed by our experimental studies, this refined TSP heuristic is much more efficient - causing fewer timeouts and improving the performance (in terms of penalized average runtime) by an order of magnitude - and thereby challenges the state of the art in TSP solving.