Computing small partial coverings

We study the generalization of covering problems such as the set cover problem to partial covering problems. Here we only want to cover a given number k of elements rather than all elements. For instance, in the k-partial (weighted) set cover problem, we wish to compute a minimum weight collection of sets that covers at least k elements. As a main result, we show that the k-partial set cover problem and its special cases like the k-partial vertex cover problem are all fixed parameter tractable (with parameter k). As a second example, we consider the minimum weight k-partial t-restricted cycle cover problem. (C) 2002 Elsevier Science B.V. All rights reserved.