IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v9y2005i3d10.1007_s10878-005-1412-9.html
   My bibliography  Save this article

The Polymatroid Steiner Problems

Author

Listed:
  • G. Calinescu

    (Illinois Institute of Technology)

  • A. Zelikovsky

    (Georgia State University)

Abstract

The Steiner tree problem asks for a minimum cost tree spanning a given set of terminals S⊂eqV in a weighted graph G = (V,E,c), c:E→ R+. In this paper we consider a generalization of the Steiner tree problem, so called Polymatroid Steiner Problem, in which a polymatroid P = P(V) is defined on V and the Steiner tree is required to span at least one base of P (in particular, there may be a single base S⊂eqV). This formulation is motivated by the following application in sensor networks – given a set of sensors S = {s1,…,s k }, each sensor s i can choose to monitor only a single target from a subset of targets X i , find minimum cost tree spanning a set of sensors capable of monitoring the set of all targets X = X1 ∪ … ∪ X k . The Polymatroid Steiner Problem generalizes many known Steiner tree problem formulations including the group and covering Steiner tree problems. We show that this problem can be solved with the polylogarithmic approximation ratio by a generalization of the combinatorial algorithm of Chekuri et al. (2002). We also define the Polymatroid directed Steiner problem which asks for a minimum cost arborescence connecting a given root to a base of a polymatroid P defined on the terminal set S. We show that this problem can be approximately solved by algorithms generalizing methods of Chekuri et al. (2002).

Suggested Citation

  • G. Calinescu & A. Zelikovsky, 2005. "The Polymatroid Steiner Problems," Journal of Combinatorial Optimization, Springer, vol. 9(3), pages 281-294, May.
  • Handle: RePEc:spr:jcomop:v:9:y:2005:i:3:d:10.1007_s10878-005-1412-9
    DOI: 10.1007/s10878-005-1412-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-005-1412-9
    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/s10878-005-1412-9?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. Wolsey, L.A., 1982. "An analysis of the greedy algorithm for the submodular set covering problem," LIDAM Reprints CORE 519, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    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. Fatemeh Navidi & Prabhanjan Kambadur & Viswanath Nagarajan, 2020. "Adaptive Submodular Ranking and Routing," Operations Research, INFORMS, vol. 68(3), pages 856-877, May.

    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. Weidong Chen & Hao Zhong & Lidong Wu & Ding-Zhu Du, 2022. "A general greedy approximation algorithm for finding minimum positive influence dominating sets in social networks," Journal of Combinatorial Optimization, Springer, vol. 44(1), pages 1-20, August.
    2. Ran, Yingli & Zhang, Ying & Zhang, Zhao, 2021. "Parallel approximation for partial set cover," Applied Mathematics and Computation, Elsevier, vol. 408(C).
    3. Dongyue Liang & Zhao Zhang & Xianliang Liu & Wei Wang & Yaolin Jiang, 2016. "Approximation algorithms for minimum weight partial connected set cover problem," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 696-712, February.
    4. Shi, Majun & Yang, Zishen & Wang, Wei, 2021. "Minimum non-submodular cover problem with applications," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    5. Chandra Chekuri & Tanmay Inamdar & Kent Quanrud & Kasturi Varadarajan & Zhao Zhang, 2022. "Algorithms for covering multiple submodular constraints and applications," Journal of Combinatorial Optimization, Springer, vol. 44(2), pages 979-1010, September.
    6. Fatemeh Navidi & Prabhanjan Kambadur & Viswanath Nagarajan, 2020. "Adaptive Submodular Ranking and Routing," Operations Research, INFORMS, vol. 68(3), pages 856-877, May.
    7. Majun Shi & Zishen Yang & Wei Wang, 2023. "Greedy guarantees for minimum submodular cost submodular/non-submodular cover problem," Journal of Combinatorial Optimization, Springer, vol. 45(1), pages 1-16, January.
    8. Dana Angluin & James Aspnes & Lev Reyzin, 2015. "Network construction with subgraph connectivity constraints," Journal of Combinatorial Optimization, Springer, vol. 29(2), pages 418-432, February.
    9. Liao, Hao & Wu, Xingtong & Wang, Bing-Hong & Wu, Xiangyang & Zhou, Mingyang, 2019. "Solving the speed and accuracy of box-covering problem in complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 954-963.
    10. Xu Zhu & Jieun Yu & Wonjun Lee & Donghyun Kim & Shan Shan & Ding-Zhu Du, 2010. "New dominating sets in social networks," Journal of Global Optimization, Springer, vol. 48(4), pages 633-642, December.
    11. Alfredo Torrico & Mohit Singh & Sebastian Pokutta & Nika Haghtalab & Joseph (Seffi) Naor & Nima Anari, 2021. "Structured Robust Submodular Maximization: Offline and Online Algorithms," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1590-1607, October.

    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:9:y:2005:i:3:d:10.1007_s10878-005-1412-9. 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.