Covering Things with Things

An abstract NP-hard covering problem is presented and xed-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 rst algorithm is deterministic and has running time O 0(kdkn). The second algorithm is also deterministic and has running time O 0(kd(k+1) + nd+1). The third is a Monte-Carlo algorithm that runs in time O 0(kd(k+1)+c2dkd(d+1)/2eb(d+1)/2cn logn) time and is correct with probability 1 − n−c. Here, the O 0 notation hides factors that are polynomial in d. These algorithms lead to xed-parameter tractable algorithms for many geometric and non-geometric covering problems. This research took place while both author were visitors at the School of Computer Science of McGill University and was partly funded by NSERC, MITACS, FCAR and CRM. yCharg e de recherches du FNRS