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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

The paper introduces a topology-aware framework for the Traveling Salesman Problem by leveraging the RTD-Lite barcode to decompose the tour-MST gap into edge-level divergence signals. It establishes a canonical Tour-MST decomposition and defines per-edge penalties, linking topological signals to classical heuristics via α-score and enabling topology-guided 2-opt/3-opt and RTDL-informed reinforcement learning. Empirical results on TSPLIB, random Euclidean instances, and heatmap-guided large-scale problems show consistent improvements in tour quality and convergence speed, validating a practical topological signal for combinatorial search. By bridging persistent-homology with learning-based solvers, the work opens a new avenue for topology-informed optimization in CO problems.

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.

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

TL;DR

The paper introduces a topology-aware framework for the Traveling Salesman Problem by leveraging the RTD-Lite barcode to decompose the tour-MST gap into edge-level divergence signals. It establishes a canonical Tour-MST decomposition and defines per-edge penalties, linking topological signals to classical heuristics via α-score and enabling topology-guided 2-opt/3-opt and RTDL-informed reinforcement learning. Empirical results on TSPLIB, random Euclidean instances, and heatmap-guided large-scale problems show consistent improvements in tour quality and convergence speed, validating a practical topological signal for combinatorial search. By bridging persistent-homology with learning-based solvers, the work opens a new avenue for topology-informed optimization in CO problems.

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.

Paper Structure

This paper contains 21 sections, 1 theorem, 6 equations, 9 figures, 9 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Topology as a missing optimisation signal.
  3. From length optimisation to topology-aware optimisation.
  4. Related Work
  5. Approach
  6. Experiments
  7. RTDL barcodes and optimal tours
  8. Improvements in 2-opt, 3-opt
  9. RTDL for heatmap-guided 2-opt
  10. RTD for Deep Q-Learning
  11. Conclusion
  12. Additional results
  13. Details of Theorem \ref{['thm:gap_decomposition']} proof
  14. Relation of RTDL bars and alpha-score
  15. Robustness Analysis of 2-opt and RTDL variants
  16. ...and 6 more sections

Key Result

Theorem 1

We construct a one-to-one correspondence $\phi$ between edges of $T$ and $T_{\text{mst}}$ so that the standard (TSP tour)-MST gap $L_{(s,t)-tour}-L_{mst}$ is decomposed as the natural sum of edge-wise non-negative topology divergence gaps: In other words, every MST edge $e$ is paired with a unique tour edge $\phi(e)$ whose weight is greater than or equal to $w(e)$, and these weight differences ex

Figures (9)

  • Figure 1: Visualization of the mapping between edges of the Hamiltonian cycle $T_{\text{tour}}$ and the minimum spanning tree $T_{\text{mst}}$. Each tour edge (except $e_{\max}$) is bijectively matched to an MST edge.
  • Figure 2: Probability of belonging to an optimal tour vs. RTDL barcode of an edge.
  • Figure 3: Avg. number of trials per iteration.
  • Figure 4: Evolution of average length during tour optimization for heatmap-guided 2-opt and 2opt + RTDL.
  • Figure 5: Example of solutions for the same TSP-500 problem through different models and optimization methods. For each model, we provide initial tour obtained through heatmap greedy decoding, 2-opt and 2-opt + RTDL optimized tours.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • proof