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Edge-wise Topological Divergence Gaps: Guiding Search in Combinatorial Optimization
arXiv:2512.15800v1 Announce Type: new
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.
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.
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