Machine Learning for Two-Stage Graph Sparsification for the Travelling Salesman Problem

High-performance TSP solvers like LKH search within a sparsified candidate graph rather than over all possible edges. Graph sparsification is non-trivial: keep too many edges and the solver wastes time; cut too many and it loses edges that belong to the optimal tour. The two leading heuristic methods, $α$-Nearest and POPMUSIC, produce high-quality candidate graphs, but no single heuristic is both sparse and reliable across all instance sizes and distributions. Machine learning methods can potentially learn better sparsification models. However, existing approaches operate on the complete graph, which is expensive and mostly restricted to Euclidean distances. To address this issue, we propose a two-stage graph sparsification approach: Stage~1 takes the union of $α$-Nearest and POPMUSIC to maximise recall; Stage~2 trains a single model to reduce density. We conducted experiments across four TSPLIB distance types, five spatial distributions, and problem sizes from 50 to 500. The two-stage approach substantially reduces candidate-graph density while retaining high coverage, generalises across distance types and distributions, outperforms recent neural sparsification methods that are restricted to Euclidean distances, and becomes increasingly valuable at larger scales where single-stage heuristics degrade.

HEATACO: Heatmap-Guided Ant Colony Decoding for Large-Scale Travelling Salesman Problems

Heatmap-based non-autoregressive solvers for large-scale Travelling Salesman Problems output dense edge-probability scores, yet final performance largely hinges on the decoder that must satisfy degree-2 constraints and form a single Hamiltonian tour. Greedy commitment can cascade into irreparable mistakes at large $N$, whereas MCTS-guided local search is accurate but compute-heavy and highly engineered. We instead treat the heatmap as a soft edge prior and cast decoding as probabilistic tour construction under feasibility constraints, where the key is to correct local mis-rankings via inexpensive global coordination. Based on this view, we introduce HeatACO, a plug-and-play Max-Min Ant System decoder whose transition policy is softly biased by the heatmap while pheromone updates provide lightweight, instance-specific feedback to resolve global conflicts; optional 2-opt/3-opt post-processing further improves tour quality. On TSP500/1K/10K, using heatmaps produced by four pretrained predictors, HeatACO+2opt achieves gaps down to 0.11%/0.23%/1.15% with seconds-to-minutes CPU decoding for fixed heatmaps, offering a better quality--time trade-off than greedy decoding and published MCTS-based decoders. Finally, we find the gains track heatmap reliability: under distribution shift, miscalibration and confidence collapse bound decoding improvements, suggesting heatmap generalisation is a primary lever for further progress.