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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
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    References listed on IDEAS

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    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).
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    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.

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