Covering things with things

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 running time. O’(kdkN).The second algorithm is also deterministic and has running time O’ (kd(k+1)+nd+1). The third is a Monte-Carlo algorithm that runs in time O’(kd(k+1) +c2dk└(d+1)/2┘└(d+1)/2┘nlogn) time and is correct with probability 1-n-c. Here, the O’ notation hides factors that are polynomial in d. These algorithms lead to ixed-parameter tractable algorithms for many geometric and non-geometric covering problems.