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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Langerman, S., Morin, P. Covering Things with Things. Discrete Comput Geom 33, 717–729 (2005). https://doi.org/10.1007/s00454-004-1108-4
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DOI: https://doi.org/10.1007/s00454-004-1108-4