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Budgeted Prize-Collecting Traveling Salesman and Minimum Spanning Tree Problems

Author

Listed:
  • Alice Paul

    (Data Science Initiative, Brown University, Providence, Rhode Island 02912;)

  • Daniel Freund

    (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02142;)

  • Aaron Ferber

    (Operations Research and Information Engineering, Cornell University, Ithaca, New York 14850)

  • David B. Shmoys

    (Operations Research and Information Engineering, Cornell University, Ithaca, New York 14850)

  • David P. Williamson

    (Operations Research and Information Engineering, Cornell University, Ithaca, New York 14850)

Abstract

We consider constrained versions of the prize-collecting traveling salesman and the prize-collecting minimum spanning tree problems. The goal is to maximize the number of vertices in the returned tour/tree subject to a bound on the tour/tree cost. Rooted variants of the problems have the additional constraint that a given vertex, the root, must be contained in the tour/tree. We present a 2-approximation algorithm for the rooted and unrooted versions of both the tree and tour variants. The algorithm is based on a parameterized primal–dual approach. It relies on first finding a threshold value for the dual variable corresponding to the budget constraint in the primal and then carefully constructing a tour/tree that is, in a precise sense, just within budget. We improve upon the best-known guarantee of 2 + ε for the rooted and unrooted tour versions and 3 + ε for the rooted and unrooted tree versions. Our analysis extends to the setting with weighted vertices, in which we want to maximize the total weight of vertices in the tour/tree. Interestingly enough, the algorithm and analysis for the rooted case and the unrooted case are almost identical.

Suggested Citation

  • Alice Paul & Daniel Freund & Aaron Ferber & David B. Shmoys & David P. Williamson, 2020. "Budgeted Prize-Collecting Traveling Salesman and Minimum Spanning Tree Problems," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 576-590, May.
    • Handle: RePEc:inm:ormoor:v:45:y:2020:i:2:p:576-590
    DOI: 10.1287/moor.2019.1002
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    References listed on IDEAS

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    1. Gerhard Reinelt, 1991. "TSPLIB—A Traveling Salesman Problem Library," INFORMS Journal on Computing, INFORMS, vol. 3(4), pages 376-384, November.
    2. Kaspi, Mor & Raviv, Tal & Tzur, Michal, 2016. "Detection of unusable bicycles in bike-sharing systems," Omega, Elsevier, vol. 65(C), pages 10-16.
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    Cited by:

    1. Ben Hermans & Roel Leus & Jannik Matuschke, 2022. "Exact and Approximation Algorithms for the Expanding Search Problem," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 281-296, January.

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