IDEAS home Printed from https://ideas.repec.org/p/cdl/itsrrp/qt4z68041r.html
   My bibliography  Save this paper

Lower and Upper Bounds for a Symmetric Multiple Depot, Multiple Travelling Salesman Problem

Author

Listed:
  • Rathinam, Sivakumar
  • Sengupta, Raja

Abstract

This paper extends the well known Held-Karp's lower bound available for a single Travelling Salesman Problem to the multiple depot case. The LP-relaxation of a symmetric multiple vehicle, multiple depot problem is shown to be lower bounded by an infinite family of bounds. Each lower bound can be computed in a tractable way using a matroid intersection algorithm. When the costs of travelling between any two locations satisfy triangle inequality, it is shown that there exists a 2-approximation algorithm for solving the multiple depot, multiple TSP. These results are useful in solving the following path planning problem of UAVs: Given a set of UAVs, their starting locations, a set of final UAV locations, a set of destinations to visit and the cost of travelling between any two locations, find a path for each UAV such that each destination is visited once by any one UAV and the total cost travelled by all the UAVs is minimum.

Suggested Citation

  • Rathinam, Sivakumar & Sengupta, Raja, 2006. "Lower and Upper Bounds for a Symmetric Multiple Depot, Multiple Travelling Salesman Problem," Institute of Transportation Studies, Research Reports, Working Papers, Proceedings qt4z68041r, Institute of Transportation Studies, UC Berkeley.
    • Handle: RePEc:cdl:itsrrp:qt4z68041r
    as

    Download full text from publisher

    File URL: https://www.escholarship.org/uc/item/4z68041r.pdf;origin=repeccitec
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Michael Held & Richard M. Karp, 1970. "The Traveling-Salesman Problem and Minimum Spanning Trees," Operations Research, INFORMS, vol. 18(6), pages 1138-1162, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Bérczi, Kristóf & Mnich, Matthias & Vincze, Roland, 2023. "Approximations for many-visits multiple traveling salesman problems," Omega, Elsevier, vol. 116(C).
    2. José Alejandro Cornejo-Acosta & Jesús García-Díaz & Julio César Pérez-Sansalvador & Carlos Segura, 2023. "Compact Integer Programs for Depot-Free Multiple Traveling Salesperson Problems," Mathematics, MDPI, vol. 11(13), pages 1-25, July.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Gábor Braun & Samuel Fiorini & Sebastian Pokutta & David Steurer, 2015. "Approximation Limits of Linear Programs (Beyond Hierarchies)," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 756-772, March.
    2. Yanling Chu & Xiaoju Zhang & Zhongzhen Yang, 2017. "Multiple quay cranes scheduling for double cycling in container terminals," PLOS ONE, Public Library of Science, vol. 12(7), pages 1-19, July.
    3. Ibrahim Muter & Tevfik Aytekin, 2017. "Incorporating Aggregate Diversity in Recommender Systems Using Scalable Optimization Approaches," INFORMS Journal on Computing, INFORMS, vol. 29(3), pages 405-421, August.
    4. Rostami, Borzou & Malucelli, Federico & Belotti, Pietro & Gualandi, Stefano, 2016. "Lower bounding procedure for the asymmetric quadratic traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 253(3), pages 584-592.
    5. Schuijbroek, J. & Hampshire, R.C. & van Hoeve, W.-J., 2017. "Inventory rebalancing and vehicle routing in bike sharing systems," European Journal of Operational Research, Elsevier, vol. 257(3), pages 992-1004.
    6. Martinhon, Carlos & Lucena, Abilio & Maculan, Nelson, 2004. "Stronger K-tree relaxations for the vehicle routing problem," European Journal of Operational Research, Elsevier, vol. 158(1), pages 56-71, October.
    7. de Klerk, E. & Pasechnik, D.V. & Sotirov, R., 2007. "On Semidefinite Programming Relaxations of the Travelling Salesman Problem (Replaced by DP 2008-96)," Discussion Paper 2007-101, Tilburg University, Center for Economic Research.
    8. Thomas L. Morin & Roy E. Marsten, 1974. "Brand-and-Bound Strategies for Dynamic Programming," Discussion Papers 106, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    9. G. Rius-Sorolla & J. Maheut & Jairo R. Coronado-Hernandez & J. P. Garcia-Sabater, 2020. "Lagrangian relaxation of the generic materials and operations planning model," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 28(1), pages 105-123, March.
    10. Ghosh, Diptesh & Sumanta Basu, 2011. "Diversified Local Search for the Traveling Salesman Problem," IIMA Working Papers WP2011-01-03, Indian Institute of Management Ahmedabad, Research and Publication Department.
    11. Kouhei Harada, 2021. "A Feasibility-Ensured Lagrangian Heuristic for General Decomposable Problems," SN Operations Research Forum, Springer, vol. 2(4), pages 1-26, December.
    12. Yao, Yu & Zhu, Xiaoning & Dong, Hongyu & Wu, Shengnan & Wu, Hailong & Carol Tong, Lu & Zhou, Xuesong, 2019. "ADMM-based problem decomposition scheme for vehicle routing problem with time windows," Transportation Research Part B: Methodological, Elsevier, vol. 129(C), pages 156-174.
    13. Horbach, Andrei, 2005. "Combinatorial relaxation of the k-traveling salesman problem," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 599, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    14. Arianna Alfieri & Shuyu Zhou & Rosario Scatamacchia & Steef L. van de Velde, 2021. "Dynamic programming algorithms and Lagrangian lower bounds for a discrete lot streaming problem in a two-machine flow shop," 4OR, Springer, vol. 19(2), pages 265-288, June.
    15. Peter Reiter & Walter Gutjahr, 2012. "Exact hybrid algorithms for solving a bi-objective vehicle routing problem," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 20(1), pages 19-43, March.
    16. Daniel Adelman, 2004. "A Price-Directed Approach to Stochastic Inventory/Routing," Operations Research, INFORMS, vol. 52(4), pages 499-514, August.
    17. Laureano Escudero, 2009. "On a mixture of the fix-and-relax coordination and Lagrangian substitution schemes for multistage stochastic mixed integer programming," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(1), pages 5-29, July.
    18. Rathinam, Sivakumar & Sengupta, Raja, 2006. "Matroid Intersection and its application to a Multiple Depot, Multiple TSP," Institute of Transportation Studies, Research Reports, Working Papers, Proceedings qt9sj6585p, Institute of Transportation Studies, UC Berkeley.
    19. Marcel Turkensteen & Dmitry Malyshev & Boris Goldengorin & Panos M. Pardalos, 2017. "The reduction of computation times of upper and lower tolerances for selected combinatorial optimization problems," Journal of Global Optimization, Springer, vol. 68(3), pages 601-622, July.
    20. P. Chardaire & G. P. McKeown & S. A. Verity-Harrison & S. B. Richardson, 2005. "Solving a Time-Space Network Formulation for the Convoy Movement Problem," Operations Research, INFORMS, vol. 53(2), pages 219-230, April.

    More about this item

    Keywords

    Modeling and Network Analysis;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:cdl:itsrrp:qt4z68041r. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Lisa Schiff (email available below). General contact details of provider: https://edirc.repec.org/data/itucbus.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.