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On interval and circular-arc covering problems

Author

Listed:
  • Reuven Cohen

    (Bar-Ilan University)

  • Mira Gonen

    (Ariel University)

Abstract

In this paper we study several related problems of finding optimal interval and circular-arc covering. We present solutions to the maximum k-interval (k-circular-arc) coverage problems, in which we want to cover maximum weight by selecting k intervals (circular-arcs) out of a given set of intervals (circular-arcs), respectively, the weighted interval covering problem, in which we want to cover maximum weight by placing k intervals with a given length, and the k-centers problem. The general sets version of the discussed problems, namely the general measure k-centers problem and the maximum covering problem for sets are known to be NP-hard. However, for the one dimensional restrictions studied here, and even for circular-arc graphs, we present efficient, polynomial time, algorithms that solve these problems. Our results for the maximum k-interval and k-circular-arc covering problems hold for any right continuous positive measure on $$\mathbb {R}$$ R .

Suggested Citation

  • Reuven Cohen & Mira Gonen, 2019. "On interval and circular-arc covering problems," Annals of Operations Research, Springer, vol. 275(2), pages 281-295, April.
    • Handle: RePEc:spr:annopr:v:275:y:2019:i:2:d:10.1007_s10479-018-3025-6
    DOI: 10.1007/s10479-018-3025-6
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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.
    3. de Werra, Dominique & Eisenbeis, Christine & Lelait, Sylvain & Stohr, Elena, 2002. "Circular-arc graph coloring: On chords and circuits in the meeting graph," European Journal of Operational Research, Elsevier, vol. 136(3), pages 483-500, February.
    4. 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.
    5. Alberto Caprara & Paolo Toth & Matteo Fischetti, 2000. "Algorithms for the Set Covering Problem," Annals of Operations Research, Springer, vol. 98(1), pages 353-371, December.
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    Cited by:

    1. Songhua Li & Minming Li & Lingjie Duan & Victor C. S. Lee, 2022. "Online algorithms for the maximum k-interval coverage problem," Journal of Combinatorial Optimization, Springer, vol. 44(5), pages 3364-3404, December.

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