## Abstract

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.

Original language | English |
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Pages (from-to) | 489-509 |

Number of pages | 21 |

Journal | Algorithmica |

Volume | 59 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2011 |

Externally published | Yes |

### Bibliographical note

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

## Keywords

- Approximation algorithms
- Lagrangian relaxation
- Partial cover

## ASJC Scopus subject areas

- Computer Science (all)
- Computer Science Applications
- Applied Mathematics