A Speed-up for Helsgaun's TSP Heuristic by Relaxing the Positive Gain Criterion

TL;DR

This work tackles the Traveling Salesman Problem by enhancing the Lin-Kernighan-Helsgaun heuristic (LKH) through relaxing the positive gain criterion that governs edge-exchange steps. It introduces two relaxations—homogeneous and tilted—that enlarge the search space of alternating circles while preserving end-of-move positive gain, yielding substantial speedups on large instances with minimal impact on solution quality. Extensive experiments across TSPLIB, VLSI, DIMACS, nationalTSP, and $T_{nm}$ benchmarks show average improvements around 10–14% in runtime, with particular gains on very large problems; the tilted relaxation consistently outperforms the homogeneous variant for large instances. The findings suggest that relaxing the positive gain criterion is a practical and fruitful direction for scaling LK-type heuristics and invites further refinements and learning-based enhancements.

Abstract

The Traveling Salesman Problem (TSP) is one of the most extensively researched and widely applied combinatorial optimization problems. It is NP-hard even in the symmetric and metric case. Building upon elaborate research, state-of-the-art exact solvers such as CONCORDE can solve TSP instances with several ten thousand vertices. A key ingredient for these integer programming approaches are fast heuristics to find a good initial solution, in particular the Lin-Kernighan-Helsgaun (LKH) heuristic. For instances with few hundred vertices heuristics like LKH often find an optimal solution. In this work we develop variations of LKH that perform significantly better on large instances. LKH repeatedly improves an initially random tour by exchanging edges along alternating circles. Thereby, it respects several criteria designed to quickly find alternating circles that give a feasible improvement of the tour. Among those criteria, the positive gain criterion stayed mostly untouched in previous research. It requires that, while constructing an alternating circle, the total gain has to be positive after each pair of edges. We relax this criterion carefully leading to improvement steps hitherto undiscovered by LKH. We confirm this improvement experimentally via extensive simulations on various benchmark libraries for TSP. Our computational study shows that for large instances our method is on average 13% faster than the latest version of LKH.

Figures (9)

• Figure 1: Running example adjusted from thesis_BO: \ref{['LKedgeweights:figure']} depicts edge costs. A possible starting tour is in \ref{['starttour']}. Example of a $2$-opt-move in \ref{['2opt']}: Edges of starting tour are black, we delete dashed red edges and insert blue edges.
• Figure 2: Display of \ref{['exampleLKHnormal']}. Black edges belong to starting tour. Red edges indicate removal. Blue edges indicate insertion. In \ref{['lkh2']} we select one of the red dotted edges to be removed, in \ref{['lkh3']} we choose one of the blue dotted edges to be added.
• Figure 3: Simplified procedure of improved LKH procedure with relaxed positive gain criterion displayed in red. Figure adjusted from thesis_BO.
• Figure 4: Display of \ref{['exampleLKHimproved']}: Relaxed positive gain criterion. Black edges belong to starting tour. Red edges indicate removal. Blue edges indicate insertion. In \ref{['lkhimproved2.2']} we choose one of the red dotted edges to be removed, in \ref{['lkhimproved2.3']} we select one of the blue dotted edges to be added.
• Figure 5: Comparison of our variant using the tilted relaxation \ref{['C2**:tiltedgain']} and the original LKH on large instances ($>30000$ vertices) in the average time per run with 5 POPMUSIC candidates. Instances that did not finish in one month of computation time are not displayed.

Theorems & Definitions (3)

• Example 2.1
• Example 3.1
• Example 4.1