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Prize-Collecting Asymmetric Traveling Salesman Problem Admits Polynomial Time Approximation Within a Constant Ratio

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Optimization and Applications (OPTIMA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13781))

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Abstract

The Prize-Collecting Traveling Salesman Problem is an extension of the classic Traveling Salesman Problem, where each node of the given graph can be skipped for some known penalty. The goal is to construct a closed walk minimizing the total transportation costs and accumulated penalties. This problem has numerous applications in operations research, including sustainable production, supply chains, and drone routing. In this paper, we propose the first approximation algorithm with constant ratio for the asymmetric version of the problem on a complete weighted digraph, where the transportation costs fulfill the triangle inequality. Employing an arbitrary \(\alpha \)-approximation algorithm for the Asymmetric Traveling Salesman Problem (ATSP) as a building block, our algorithm establishes an \((\alpha +2)\)-approximation for the Prize-Collecting Asymmetric Traveling Salesman Problem. In particular, using the seminal recent Swensson-Traub \((22+\varepsilon )\)-approximation algorithm for the ATSP, we obtain \((24+\varepsilon )\)-approximate solutions for our problem.

This research was carried out under the financial support of the Russian Science Foundation, grant no. 22-21-00672, https://rscf.ru/project/22-21-00672/.

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Authors and Affiliations

  1. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay, Katherine Neznakhina & Ksenia Rizhenko

  2. Ural Federal University, Ekaterinburg, Russia

    Katherine Neznakhina

Authors
  1. Michael Khachay
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  2. Katherine Neznakhina
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  3. Ksenia Rizhenko
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Correspondence to Michael Khachay .

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Editors and Affiliations

  1. Dorodnitsyn Computing Centre, Moscow, Russia

    Nicholas Olenev

  2. Dorodnitsyn Computing Centre, Moscow, Russia

    Yuri Evtushenko

  3. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  4. Krasovskii Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  5. Dorodnitsyn Computing Centre, Moscow, Russia

    Vlasta Malkova

  6. Dorodnitsyn Computing Centre, Moscow, Russia

    Igor Pospelov

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Khachay, M., Neznakhina, K., Rizhenko, K. (2022). Prize-Collecting Asymmetric Traveling Salesman Problem Admits Polynomial Time Approximation Within a Constant Ratio. In: Olenev, N., Evtushenko, Y., Jaćimović, M., Khachay, M., Malkova, V., Pospelov, I. (eds) Optimization and Applications. OPTIMA 2022. Lecture Notes in Computer Science, vol 13781. Springer, Cham. https://doi.org/10.1007/978-3-031-22543-7_6

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  • DOI: https://doi.org/10.1007/978-3-031-22543-7_6

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