Discrete & Computational Geometry

, Volume 33, Issue 4, pp 717–729 | Cite as

Covering Things with Things

Article

Abstract

An abstract NP-hard covering problem is presented and fixed-parameter tractable algorithms for this problem are described. The running times of the algorithms are expressed in terms of three parameters: $n$, the number of elements to be covered, $k$, the number of sets allowed in the covering, and $d$, the combinatorial dimension of the problem. The first algorithm is deterministic and has a running time of $O’(k^{dk}n)$. The second algorithm is also deterministic and has a running time of $O’(k^{d(k+1)}+n^{d+1})$. The third is a Monte-Carlo algorithm that runs in time $O’(\runtime)$ and is correct with probability $1-n^{-c}$. Here, the $O’$ notation hides factors that are polynomial in $d$. These algorithms lead to fixed-parameter tractable algorithms for many geometric and non-geometric covering problems.

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Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Département d’informatique, Université Libre de Bruxelles, Avenue Franklin D. Roosevelt 50, B-1050 BruxellesBelgium
  2. 2.School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6Canada

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