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

TL;DR

HeatACO reframes heatmap-to-tour decoding for large-scale TSP as probabilistic constrained sampling guided by a soft edge prior. By embedding the heatmap in a Max--Min Ant System and applying lightweight, instance-specific pheromone feedback, HeatACO achieves strong quality–time trade-offs with optional 2-opt/3-opt refinement on problems up to $N=10{,}000$ and across multiple heatmap predictors. The approach outperforms greedy decoding and matches or surpasses published MCTS-based decoders within practical CPU budgets, while offering robust performance when heatmaps preserve high-recall confidence bands. The study also highlights the central role of heatmap reliability and discusses calibration and distribution-shift challenges as levers for further improvements in heatmap-guided decoding. Overall, HeatACO provides a scalable, plug-and-play decoding solution that shifts computational burden toward the decoding stage and clarifies the impact of heatmap quality on real-world routing applications.

Abstract

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.

Figures (10)

• Figure 1: Overview of heatmap-based decoding. Left: heatmap prediction (NAR), producing an edge-confidence matrix $H$ from node coordinates. Top-right: standard ACO decoding constructs a tour using distance heuristics and pheromone feedback. Bottom-right:HeatACO injects $H$ as a soft prior into ACO/MMAS sampling while preserving pheromone feedback to correct globally inconsistent local choices.
• Figure 2: Convergence without local improvement across scales. Let $L_i(t)$ be the mean best-so-far tour length at iteration $t$ for method $i$ (averaged over instances and 10 seeds). For each panel we define a common final baseline $B=\min_i L_i(I)$ at the last iteration $I$, then plot $y_i(t)=\log_{10}(L_i(t)-B+c)$ with $c=\max(\alpha\,\mathrm{median}_i(L_i(t_{\text{mid}})-B),\epsilon)$, $t_{\text{mid}}=0.5I$, $\alpha=0.03$, and $\epsilon=\max(|B|\cdot 10^{-8},10^{-12})$. This highlights mid/late-iteration differences without over-amplifying tiny tail gaps. Appendix \ref{['app:convergence']} reports the corresponding curves with 2-opt local improvement.
• Figure 3: Interval contribution analysis of heatmap confidence values across instance scales. We partition heatmap scores into confidence intervals on $[0,1]$. Bars report the (normalised) number of candidate edges per node whose confidence falls in each interval. The curve is normalised over the optimal-tour edges: for each interval, it reports the fraction of optimal-tour edges whose confidence falls in that interval (so the curve sums to 1 across intervals). Benchmark-tour edges concentrate in a narrow mid/high-confidence band, while the low-confidence tail contains many candidate edges but very few optimal-tour edges.
• Figure 4: Sensitivity of HeatACO (no local search) to the heatmap exponent $\gamma$ across instance scales. Each cell reports the optimality gap (%) achieved by HeatACO when using a fixed heatmap source and a coarse $\gamma\in\{0.1,0.5,1.0,2.0\}$ sweep.
• Figure 5: Decoding gap versus auxiliary heatmap cross-entropy diagnostics on TSP500/1K/10K benchmarks. Lines show the optimality gap (%) achieved by different decoders for each heatmap source; bars show the corresponding CE and WCE values from Table \ref{['tab:coverage']} (normalised per instance scale for visualisation). In the legend, "NAR+ACO" denotes HeatACO (heatmap-guided MMAS), and "+2opt/+3opt" denote adding local improvement.