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On nonlinear multi-covering problems

Author

Listed:
  • Reuven Cohen

    (Bar-Ilan University)

  • Mira Gonen

    (Ariel University)

  • Asaf Levin

    (The Technion)

  • Shmuel Onn

    (The Technion)

Abstract

In this paper we define the exact k-coverage problem, and study it for the special cases of intervals and circular-arcs. Given a set system consisting of a ground set of n points with integer demands $$\{d_0,\dots ,d_{n-1}\}$$ { d 0 , ⋯ , d n - 1 } and integer rewards, subsets of points, and an integer k, select up to k subsets such that the sum of rewards of the covered points is maximized, where point i is covered if exactly $$d_i$$ d i subsets containing it are selected. Here we study this problem and some related optimization problems. We prove that the exact k-coverage problem with unbounded demands is NP-hard even for intervals on the real line and unit rewards. Our NP-hardness proof uses instances where some of the natural parameters of the problem are unbounded (each of these parameters is linear in the number of points). We show that this property is essential, as if we restrict (at least) one of these parameters to be a constant, then the problem is polynomial time solvable. Our polynomial time algorithms are given for various generalizations of the problem (in the setting where one of the parameters is a constant).

Suggested Citation

  • Reuven Cohen & Mira Gonen & Asaf Levin & Shmuel Onn, 2017. "On nonlinear multi-covering problems," Journal of Combinatorial Optimization, Springer, vol. 33(2), pages 645-659, February.
    • Handle: RePEc:spr:jcomop:v:33:y:2017:i:2:d:10.1007_s10878-015-9985-4
    DOI: 10.1007/s10878-015-9985-4
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    References listed on IDEAS

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    1. CORNUEJOLS, Gérard & EMHAUSER, George L. & WOLSEY, Laurence A., 1980. "Worst-case and probabilistic analysis of algorithms for a location problem," LIDAM Reprints CORE 437, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Gerard Cornuejols & George L. Nemhauser & Laurence A. Wolsey, 1980. "Worst-Case and Probabilistic Analysis of Algorithms for a Location Problem," Operations Research, INFORMS, vol. 28(4), pages 847-858, August.
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    Cited by:

    1. Reuven Cohen & Mira Gonen, 2019. "On interval and circular-arc covering problems," Annals of Operations Research, Springer, vol. 275(2), pages 281-295, April.

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