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

TL;DR

The paper investigates how candidate-edge lists influence the Lin-Kernighan-Helsgaun (LKH) TSP heuristic, identifying fixed candidate sets as a key source of timeouts. It compares α-candidate, 2-opt, and POPMUSIC-based candidate sets, introducing a restartable LKH variant that uses POPMUSIC (LKH_pop_restart). Empirical results on 10,000 evolved instances show that 2-opt candidate sets dramatically reduce timeouts and that alternating candidate sets via restarts with POPMUSIC yields the best overall performance, challenging the state-of-the-art EAX in many settings. The work highlights the practical impact of candidate-set design on TSP solving and suggests directions for algorithm selection and further optimization of candidate construction.

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.

Figures (9)

• Figure 1: A base minimum 1-tree is composed of an MST (gray edges), along with two additional (blue dashed) edges connecting the excluded node with its first and second nearest neighbors.
• Figure 2: The minimum 1-tree required to include an additional (green dashed-dotted) edge does not have to include the second nearest neighbor edge (red dotted line). Thus, the $\alpha$-nearness value of the new edge is the difference between itself and the red edge.
• Figure 3: A minimalistic example of a TSP instance (with five nodes) that is not solvable by LKH. Left: The corresponding $\alpha$-candidate set. Right: The initial tour (solid red lines) with two missing edges from the optimal tour (blue dashed lines).
• Figure 4: $\alpha$-candidate set (left) and combined edges of $1,\!000$ 2-opt runs (right) with the optimal tour for tsplib instance d2103. Gray edges are not in the optimal tour, while black edges are part of it. Red edges are optimal, but missing in the respective candidate set.
• Figure 5: Share of problem instances with $0$ to $18$ missing edges in the $\alpha$-candidate list (x-axis) for the different groups and instance sizes.