TY - JOUR

T1 - A unified approach to approximating partial covering problems

AU - Könemann, Jochen

AU - Parekh, Ojas

AU - Segev, Danny

N1 - Funding Information:
Research of J. Könemann was supported by NSERC grant no. 288340-2004.

PY - 2011/4

Y1 - 2011/4

N2 - An instance of the generalized partial cover problem consists of a ground set U and a family of subsets S ⊆ 2 U. Each element e ∈ U is associated with a profit p(e), whereas each subset S ⊆ S has a cost c(S). The objective is to find a minimum cost subcollection S′ ⊆ S such that the combined profit of the elements covered by S′ is at least P, a specified profit bound. In the prize-collecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e U uncovered, we incur a penalty of π(e). The goal is to identify a subcollection S′ ⊆ S that minimizes the cost of S′ plus the penalties of uncovered elements. Although problem-specific connections between the partial cover and the prize-collecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first non-trivial approximability results.

AB - An instance of the generalized partial cover problem consists of a ground set U and a family of subsets S ⊆ 2 U. Each element e ∈ U is associated with a profit p(e), whereas each subset S ⊆ S has a cost c(S). The objective is to find a minimum cost subcollection S′ ⊆ S such that the combined profit of the elements covered by S′ is at least P, a specified profit bound. In the prize-collecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e U uncovered, we incur a penalty of π(e). The goal is to identify a subcollection S′ ⊆ S that minimizes the cost of S′ plus the penalties of uncovered elements. Although problem-specific connections between the partial cover and the prize-collecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first non-trivial approximability results.

KW - Approximation algorithms

KW - Lagrangian relaxation

KW - Partial cover

UR - http://www.scopus.com/inward/record.url?scp=80054710577&partnerID=8YFLogxK

U2 - 10.1007/s00453-009-9317-0

DO - 10.1007/s00453-009-9317-0

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AN - SCOPUS:80054710577

SN - 0178-4617

VL - 59

SP - 489

EP - 509

JO - Algorithmica

JF - Algorithmica

IS - 4

ER -