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A better-than-1.6-approximation for prize-collecting TSP.

Jannis Blauth1, Nathan Klein2, Martin Nägele1

  • 1Department of Mathematics, ETH Zurich, Zurich, Switzerland.

Mathematical Programming
|May 1, 2026
PubMed
Summary
This summary is machine-generated.

This study introduces a new approximation algorithm for Prize-Collecting TSP, improving the solution quality guarantee to 1.599. The algorithm enhances efficiency for the Prize-Collecting Stroll variant as well.

Keywords:
Apprioximation AlgorithmsCombinatorial optimizationTSP

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Area of Science:

  • Operations Research
  • Computer Science
  • Combinatorial Optimization

Background:

  • The Prize-Collecting Traveling Salesperson Problem (TSP) involves minimizing tour length plus penalties for unvisited vertices.
  • Existing approximation algorithms for Prize-Collecting TSP have limitations in solution quality.

Purpose of the Study:

  • To develop a more effective approximation algorithm for the Prize-Collecting TSP.
  • To improve the approximation ratio compared to existing methods.

Main Methods:

  • The study utilizes a polynomial-time approximation algorithm based on linear programming relaxation.
  • The approach involves decomposing solutions into rooted trees, followed by pruning and parity correction.
  • The algorithm's performance is analyzed using mathematical bounds, including the golden ratio.

Main Results:

  • The proposed algorithm achieves an approximation guarantee of approximately 1.599 for Prize-Collecting TSP, surpassing the previous 1.774 ratio.
  • For the Prize-Collecting Stroll (path version), an improved approximation guarantee of 1.6662 is demonstrated, outperforming the prior 1.926 ratio.

Conclusions:

  • The new algorithm offers a significant improvement in approximation guarantees for Prize-Collecting TSP and its path variant.
  • This research advances the field of combinatorial optimization by providing more efficient solutions for complex routing problems.