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A Unified Approach to Approximating Partial Covering Problems

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Abstract

An instance of the generalized partial cover problem consists of a ground set U and a family of subsets \({\mathcal{S}}\subseteq 2^{U}\) . Each element eU is associated with a profit p(e), whereas each subset \(S\in \mathcal{S}\) has a cost c(S). The objective is to find a minimum cost subcollection \(\mathcal{S}'\subseteq \mathcal{S}\) such that the combined profit of the elements covered by \(\mathcal{S}'\) 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 eU uncovered, we incur a penalty of π(e). The goal is to identify a subcollection \(\mathcal{S}'\subseteq \mathcal{S}\) that minimizes the cost of \(\mathcal{S}'\) 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.

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Author information

Correspondence to Danny Segev.

Additional information

An extended abstract of this paper appeared in Proceedings of the 14th Annual European Symposium on Algorithms, 2006.

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

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Könemann, J., Parekh, O. & Segev, D. A Unified Approach to Approximating Partial Covering Problems. Algorithmica 59, 489–509 (2011). https://doi.org/10.1007/s00453-009-9317-0

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Keywords

  • Partial cover
  • Approximation algorithms
  • Lagrangian relaxation