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A Branch-And-Bound Algorithm for the Traveling Salesman Problem with Difficult Neighborhoods

Authors Sándor P. Fekete , Rouven Kniep , Dominik Krupke , Michael Perk

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LIPIcs.SoCG.2026.46.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Discrete optimization
  • General and reference → Experimentation
Keywords
  • Geometric optimization
  • geometric covering
  • TSP with neighborhoods
  • exact algorithms
  • algorithm engineering

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Abstract

The Traveling Salesman Problem with Neighborhoods (TSPN) generalizes the classical Traveling Salesman Problem (TSP) by requiring a tour to visit a set of polygonal regions rather than fixed points, a natural goal that arises in various applications. While the geometric TSP allows arbitrarily close approximation and provably optimal solutions for benchmark instances of significant size, the TSPN is considerably more challenging, both in theory (due to APX-hardness) and practice, for which only benchmark instances up to 16 regions have been solved to optimality.
Here we present a branch-and-bound algorithm that combines a spectrum of geometry-based filters (for reducing the number of considered sequences) with Second-Order Cone Programs (SOCP) (for computing optimal tours for a given permutation of neighborhoods). This allows us to solve larger polygonal TSPN instances than before to within an optimality tolerance of 0.1%; moreover, while previous work (both in theory and practice) relied on relatively benign neighborhoods, we can handle non-convex, non-simple neighborhoods of different sizes. In experiments on 490 benchmark instances with up to 50 polygons each, our method achieves a 99.6% optimality rate within 300s, with the remaining two instances solved within 595s. For 68 larger instances of size n = 60, our method still allows solving 86.8% of instances to optimality within 900s, leaving only 3 of the instances with optimality gaps above 3%, with the maximum being 5.53%.

Cite As Get BibTex

Sándor P. Fekete, Rouven Kniep, Dominik Krupke, and Michael Perk. A Branch-And-Bound Algorithm for the Traveling Salesman Problem with Difficult Neighborhoods. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.46

Author Details

Sándor P. Fekete
  • Department of Computer Science, TU Braunschweig, Germany
  • L3S Research Center, Hannover, Germany
Rouven Kniep
  • Department of Computer Science, TU Braunschweig, Germany
Dominik Krupke
  • Department of Computer Science, TU Braunschweig, Germany
Michael Perk
  • Department of Computer Science, TU Braunschweig, Germany

Funding

Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) as part of project Computational Geometry: Solving Hard Optimization Problems (CG:SHOP) - 444569951.

Acknowledgements

We thank Barak Ugav for his contributions to an earlier version of the branch-and-bound implementation for the CE-TSP, on which parts of the current code are based.

Supplementary Materials

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