TY - JOUR

T1 - On nonlinear multi-covering problems

AU - Cohen, Reuven

AU - Gonen, Mira

AU - Levin, Asaf

AU - Onn, Shmuel

N1 - Publisher Copyright:
© 2015, Springer Science+Business Media New York.

PY - 2017/2/1

Y1 - 2017/2/1

N2 - 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 { d0, ⋯ , dn - 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 di 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).

AB - 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 { d0, ⋯ , dn - 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 di 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).

KW - Circular-arcs

KW - Covering problems

KW - Intervals

KW - Nonlinear integer programming

UR - http://www.scopus.com/inward/record.url?scp=84952645447&partnerID=8YFLogxK

U2 - 10.1007/s10878-015-9985-4

DO - 10.1007/s10878-015-9985-4

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AN - SCOPUS:84952645447

SN - 1382-6905

VL - 33

SP - 645

EP - 659

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

IS - 2

ER -