Edge-wise Topological Divergence Gaps: Guiding Search in Combinatorial Optimization
Ilya Trofimov, Daria Voronkova, Alexander Mironenko, Anton Dmitriev, Eduard Tulchinskii, Evgeny Burnaev, Serguei Barannikov
TL;DR
This paper introduces a topological feedback mechanism based on divergence analysis between tours and minimum spanning trees to enhance heuristics for solving the Traveling Salesman Problem, leading to improved performance and convergence.
Contribution
It presents a canonical decomposition theorem linking tour-MST gaps to topological divergence, guiding heuristic optimization in TSP.
Findings
Topology-guided optimization improves heuristic performance.
Faster convergence in tour optimization.
Effective across various instance types.
Abstract
We introduce a topological feedback mechanism for the Travelling Salesman Problem (TSP) by analyzing the divergence between a tour and the minimum spanning tree (MST). Our key contribution is a canonical decomposition theorem that expresses the tour-MST gap as edge-wise topology-divergence gaps from the RTD-Lite barcode. Based on this, we develop a topological guidance for 2-opt and 3-opt heuristics that increases their performance. We carry out experiments with fine-optimization of tours obtained from heatmap-based methods, TSPLIB, and random instances. Experiments demonstrate the topology-guided optimization results in better performance and faster convergence in many cases.
Peer Reviews
Decision·Submitted to ICLR 2026
- The proposed method can be integrated with numerical and learning based solvers - Experimental results show that the proposed method provides faster convergence and shorter tours.
- The core optimization idea is not fundamentally new. The paper’s edge badness score closely parallels the α-score in LKH-3, which already uses MST/1-tree structure to rank edges for removal. - The paper doesn't compare to state of the art TSP solvers like Concorde, LKH. - The RL experiments are limited to relatively small problem sizes (100 nodes). It is unclear whether the RTDL-shaped reward remains stable and beneficial for larger tours.
Local search and other heuristics for the TSP are an interesting area of research that has been an active field for decades. Hence, the line of research is interesting and well-motivated. Using a topological measure to speed up these heuristics is interesting. It is also interesting that this measure can be used in different heuristics (local search and Q-learning).
Please see the text in the field "Questions" below for a detailed discussion of the weaknesses. In summary, the quality of the writing could be improved and more experiments are necessary in my opinion.
1. The decomposition theorem provides a mathematically grounded link between graph topology and optimization performance. 2. Edge-wise topological divergence gives interpretable insight into why certain edges are suboptimal.
1. Although the authors claim broader applicability to combinatorial optimization, all experiments focus solely on TSP. 2. Although an ablation is included, the effect of RTDL frequency, edge selection granularity, and reward scaling could be analyzed more systematically.
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Taxonomy
TopicsVehicle Routing Optimization Methods · Metaheuristic Optimization Algorithms Research · Constraint Satisfaction and Optimization