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On interval and circular-arc covering problems

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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 \(\mathbb {R}\).

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Notes

  1. Notice that this replacement may lead to an intersection with intervals to the left. However, this does not affect the argument, as there still exists a better cover, possibly for a different t.

  2. Recall that A(a, b) is a set of a intervals that covers the maximum possible weight of points out of the leftmost b points.

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Acknowledgements

Reuven Cohen thanks the BSF for support. Science and Technology of Israel.

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Authors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, 52900, Ramat Gan, Israel

    Reuven Cohen

  2. Department of Computer Science, Ariel University, Ariel, Israel

    Mira Gonen

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  1. Reuven Cohen
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  2. Mira Gonen
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Correspondence to Mira Gonen.

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Cohen, R., Gonen, M. On interval and circular-arc covering problems. Ann Oper Res 275, 281–295 (2019). https://doi.org/10.1007/s10479-018-3025-6

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