Edge-wise Topological Divergence Gaps: Guiding Search in Combinatorial Optimization

2025-12-16

AI Summary

To address the tendency of local search methods to stagnate at local optima in the Traveling Salesman Problem (TSP), this paper introduces an edge-level feedback mechanism grounded in topological discrepancies between feasible tours and their corresponding Minimum Spanning Trees (MSTs). We establish, for the first time, a theorem decomposing the tour–MST gap at the edge level, and design RTD-Lite—a barcode-based metric quantifying local topological deviation—enabling geometry-aware, interpretable, and tunable optimization guidance. The method integrates persistent homology, graph-theoretic topology analysis, classical 2-opt/3-opt local search, and heatmap-informed initial solution generation. Evaluated on TSPLIB benchmarks, random instances, and heatmap-initialized problems, our approach achieves significantly faster convergence and higher solution quality. Experimental results validate both the effectiveness and generalizability of topological guidance in combinatorial optimization.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Topic discovery and tracking

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.

Problem

• Develops topological feedback for TSP using tour-MST divergence
• Provides canonical decomposition via edge-wise topology-divergence gaps
• Guides 2-opt and 3-opt heuristics to improve performance

Innovation

• Topological feedback mechanism using tour-MST divergence
• Canonical decomposition theorem for edge-wise topology-divergence gaps
• Topological guidance for 2-opt and 3-opt heuristics