Partial vs. Complete Domination: t-Dominating Set

We examine the parameterized complexity of t -Dominating Set , the problem of finding a set of at most knodes that dominate at least tnodes of a graph G= (V,E). The classic NP-complete problem Dominating Set , which can be seen to be t -Dominating Set with the restriction that t= n, has long been known to be W[2]-complete when parameterized in k. Whereas this implies W[2]-hardness for t -Dominating Set and the parameter k, we are able to prove fixed-parameter tractability for t -Dominating Set and the parameter t. More precisely, we obtain a quintic problem kernel and a randomized $O((4+\varepsilon)^t\textit{poly}(n))$ algorithm. The algorithm is based on the divide-and-color method introduced to the community earlier this year, rather intuitive and can be derandomized using a standard framework.