IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/vyid10.1007_s10878-019-00418-w.html
   My bibliography  Save this article

Algorithms for the metric ring star problem with fixed edge-cost ratio

Author

Listed:
  • Xujin Chen

    (Chinese Academy of Sciences
    University of Chinese Academy of Sciences)

  • Xiaodong Hu

    (Chinese Academy of Sciences
    University of Chinese Academy of Sciences)

  • Xiaohua Jia

    (City University of Hong Kong)

  • Zhongzheng Tang

    (Chinese Academy of Sciences
    University of Chinese Academy of Sciences
    City University of Hong Kong)

  • Chenhao Wang

    (Chinese Academy of Sciences
    University of Chinese Academy of Sciences
    City University of Hong Kong)

  • Ying Zhang

    (Beijing Electronic Science and Technology Institute)

Abstract

We address the metric ring star problem with fixed edge-cost ratio, abbreviated as RSP. Given a complete graph $$G=(V,E)$$ G = ( V , E ) with a specified depot node $$d\in V$$ d ∈ V , a nonnegative cost function $$c\in \mathbb {R}_+^E$$ c ∈ R + E on E which satisfies the triangle inequality, and an edge-cost ratio $$M\ge 1$$ M ≥ 1 , the RSP is to locate a ring $$R=(V',E')$$ R = ( V ′ , E ′ ) in G, a simple cycle through d or d itself, aiming to minimize the sum of two costs: the cost for constructing ring R, i.e., $$M\cdot \sum _{e\in E'}c(e)$$ M · ∑ e ∈ E ′ c ( e ) , and the cost for attaching nodes in $$V{\setminus } V'$$ V \ V ′ to their closest ring nodes (in R), i.e., $$\sum _{u\in V{\setminus } V'}\min _{v\in V'}c(uv)$$ ∑ u ∈ V \ V ′ min v ∈ V ′ c ( u v ) . We show that the singleton ring d is an optimal solution of the RSP when $$M\ge (|V|-1)/2$$ M ≥ ( | V | - 1 ) / 2 . This particularly implies a $$\sqrt{|V|-1}$$ | V | - 1 -approximation algorithm for the RSP with any $$M\ge 1$$ M ≥ 1 . We present randomized 3-approximation algorithm and deterministic 5.06-approximation algorithm for the RSP, by adapting algorithms for the tour-connected facility location problem (tour-CFLP) and single-source rent-or-buy problem due to Eisenbrand et al. and Williamson and van Zuylen, respectively. Building on the PTAS of Eisenbrand et al. for the tour-CFLP, we give a PTAS for the RSP with $$|V|/M=O(1)$$ | V | / M = O ( 1 ) . We also consider the capacitated RSP (CRSP) which puts an upper limit k on the number of leaf nodes that a ring node can serve, and present a $$(10+6M/k)$$ ( 10 + 6 M / k ) -approximation algorithm for this capacitated generalization. Heuristics based on some natural strategies are proposed for both the RSP and CRSP. Simulation results demonstrate that the proposed approximation and heuristic algorithms have good practical performances.

Suggested Citation

  • Xujin Chen & Xiaodong Hu & Xiaohua Jia & Zhongzheng Tang & Chenhao Wang & Ying Zhang, 0. "Algorithms for the metric ring star problem with fixed edge-cost ratio," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-25.
  • Handle: RePEc:spr:jcomop:v::y::i::d:10.1007_s10878-019-00418-w
    DOI: 10.1007/s10878-019-00418-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-019-00418-w
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10878-019-00418-w?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Morton Klein, 1967. "A Primal Method for Minimal Cost Flows with Applications to the Assignment and Transportation Problems," Management Science, INFORMS, vol. 14(3), pages 205-220, November.
    2. Calvete, Herminia I. & Galé, Carmen & Iranzo, José A., 2016. "MEALS: A multiobjective evolutionary algorithm with local search for solving the bi-objective ring star problem," European Journal of Operational Research, Elsevier, vol. 250(2), pages 377-388.
    3. Gerhard Reinelt, 1991. "TSPLIB—A Traveling Salesman Problem Library," INFORMS Journal on Computing, INFORMS, vol. 3(4), pages 376-384, November.
    4. Calvete, Herminia I. & Galé, Carmen & Iranzo, José A., 2013. "An efficient evolutionary algorithm for the ring star problem," European Journal of Operational Research, Elsevier, vol. 231(1), pages 22-33.
    5. J. Beasley & E. Nascimento, 1996. "The Vehicle Routing-Allocation Problem: A unifying framework," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 4(1), pages 65-86, June.
    6. Moreno Perez, Jose A. & Marcos Moreno-Vega, J. & Rodriguez Martin, Inmaculada, 2003. "Variable neighborhood tabu search and its application to the median cycle problem," European Journal of Operational Research, Elsevier, vol. 151(2), pages 365-378, December.
    7. Baldacci, R. & Dell'Amico, M., 2010. "Heuristic algorithms for the multi-depot ring-star problem," European Journal of Operational Research, Elsevier, vol. 203(1), pages 270-281, May.
    8. G. A. Croes, 1958. "A Method for Solving Traveling-Salesman Problems," Operations Research, INFORMS, vol. 6(6), pages 791-812, December.
    9. Current, John R. & Schilling, David A., 1994. "The median tour and maximal covering tour problems: Formulations and heuristics," European Journal of Operational Research, Elsevier, vol. 73(1), pages 114-126, February.
    10. R. Baldacci & M. Dell'Amico & J. Salazar González, 2007. "The Capacitated m -Ring-Star Problem," Operations Research, INFORMS, vol. 55(6), pages 1147-1162, December.
    Full references (including those not matched with items on IDEAS)

    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. Xujin Chen & Xiaodong Hu & Xiaohua Jia & Zhongzheng Tang & Chenhao Wang & Ying Zhang, 2021. "Algorithms for the metric ring star problem with fixed edge-cost ratio," Journal of Combinatorial Optimization, Springer, vol. 42(3), pages 499-523, October.
    2. Glock, Katharina & Meyer, Anne, 2023. "Spatial coverage in routing and path planning problems," European Journal of Operational Research, Elsevier, vol. 305(1), pages 1-20.
    3. Baldacci, Roberto & Hill, Alessandro & Hoshino, Edna A. & Lim, Andrew, 2017. "Pricing strategies for capacitated ring-star problems based on dynamic programming algorithms," European Journal of Operational Research, Elsevier, vol. 262(3), pages 879-893.
    4. Afsaneh Amiri & Majid Salari, 2019. "Time-constrained maximal covering routing problem," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 41(2), pages 415-468, June.
    5. Baldacci, Roberto & Hoshino, Edna A. & Hill, Alessandro, 2023. "New pricing strategies and an effective exact solution framework for profit-oriented ring arborescence problems," European Journal of Operational Research, Elsevier, vol. 307(2), pages 538-553.
    6. L Vogt & C A Poojari & J E Beasley, 2007. "A tabu search algorithm for the single vehicle routing allocation problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 58(4), pages 467-480, April.
    7. Fatih Rahim & Canan Sepil, 2014. "A location-routing problem in glass recycling," Annals of Operations Research, Springer, vol. 223(1), pages 329-353, December.
    8. Zang, Xiaoning & Jiang, Li & Liang, Changyong & Fang, Xiang, 2023. "Coordinated home and locker deliveries: An exact approach for the urban delivery problem with conflicting time windows," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 177(C).
    9. Oya Ekin Karaşan & A. Ridha Mahjoub & Onur Özkök & Hande Yaman, 2014. "Survivability in Hierarchical Telecommunications Networks Under Dual Homing," INFORMS Journal on Computing, INFORMS, vol. 26(1), pages 1-15, February.
    10. A. S. Santos & A. M. Madureira & M. L. R. Varela, 2018. "The Influence of Problem Specific Neighborhood Structures in Metaheuristics Performance," Journal of Mathematics, Hindawi, vol. 2018, pages 1-14, July.
    11. Anupam Mukherjee & Partha Sarathi Barma & Joydeep Dutta & Goutam Panigrahi & Samarjit Kar & Manoranjan Maiti, 2022. "A multi-objective antlion optimizer for the ring tree problem with secondary sub-depots," Operational Research, Springer, vol. 22(3), pages 1813-1851, July.
    12. R. Baldacci & M. Dell'Amico & J. Salazar González, 2007. "The Capacitated m -Ring-Star Problem," Operations Research, INFORMS, vol. 55(6), pages 1147-1162, December.
    13. Sheldon H. Jacobson & Shane N. Hall & Laura A. McLay & Jeffrey E. Orosz, 2005. "Performance Analysis of Cyclical Simulated Annealing Algorithms," Methodology and Computing in Applied Probability, Springer, vol. 7(2), pages 183-201, June.
    14. Lucas García & Pedro M. Talaván & Javier Yáñez, 2022. "The 2-opt behavior of the Hopfield Network applied to the TSP," Operational Research, Springer, vol. 22(2), pages 1127-1155, April.
    15. Reihaneh, Mohammad & Ghoniem, Ahmed, 2019. "A branch-and-price algorithm for a vehicle routing with demand allocation problem," European Journal of Operational Research, Elsevier, vol. 272(2), pages 523-538.
    16. Chris Walshaw, 2002. "A Multilevel Approach to the Travelling Salesman Problem," Operations Research, INFORMS, vol. 50(5), pages 862-877, October.
    17. S Salhi & A Al-Khedhairi, 2010. "Integrating heuristic information into exact methods: The case of the vertex p-centre problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(11), pages 1619-1631, November.
    18. Ivan Contreras & Moayad Tanash & Navneet Vidyarthi, 2017. "Exact and heuristic approaches for the cycle hub location problem," Annals of Operations Research, Springer, vol. 258(2), pages 655-677, November.
    19. Rafael Blanquero & Emilio Carrizosa & Amaya Nogales-Gómez & Frank Plastria, 2014. "Single-facility huff location problems on networks," Annals of Operations Research, Springer, vol. 222(1), pages 175-195, November.
    20. Martins, Francisco Leonardo Bezerra & do Nascimento, José Cláudio, 2022. "Power law dynamics in genealogical graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 596(C).

    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:spr:jcomop:v::y::i::d:10.1007_s10878-019-00418-w. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.