Hitting and Covering Partially

TL;DR: This paper designs a kernel for Partial \(d\)-Exact SC in which sizes of the universe and the family are bounded by \({\mathcal {O}}(k^{d+1})\), and studies variants of d-Hitting Set and d-Set Cover, which are called Partial d -Hitting set (Partial \(d)-HS) and Partial \( d-Exact Set Cover (Partic \(d-Exacts SC), respectively.

Abstract: d-Hitting Set and d-Set Cover are among the classical NP-hard problems. In this paper, we study variants of d-Hitting Set and d-Set Cover, which are called Partial d -Hitting Set (Partial \(d\)-HS) and Partial \(d\)-Exact Set Cover (Partial \(d\)-Exact SC), respectively. In Partial \(d\)-HS, given a universe U, a family \({\mathcal F}\), of sets of size at most d over U, and integers k and t, the objective is to decide if there exists a \(S \subseteq U\) of size at most k such that S intersects with at least t sets in \(\mathcal {F}\). We obtain a kernel for Partial \(d\)-HS in which the size of the universe is bounded by \({\mathcal {O}}(dt)\) and the size of the family is bounded by \({\mathcal {O}}(dt^2)\). Using this result, we obtain a kernel for Partial Vertex Cover (PVC) with \({\mathcal {O}}(t)\) vertices, where t is the number of edges to be covered. Next, we study the Partial \(d\)-Exact SC problem, where, given a universe U, a family \({\mathcal F}\), of sets of size exactly d over U, and integers k and t, the objective is to decide if there is \({\mathcal S}\subseteq {\mathcal F}\) of size at most k, such that \({\mathcal S}\) covers at least t elements in U. We design a kernel for Partial \(d\)-Exact SC in which sizes of the universe and the family are bounded by \({\mathcal {O}}(k^{d+1})\). Finally, we study a special case of Partial \(d\)-HS, when \(d=2\), and design an exact exponential time algorithm with running time \({\mathcal {O}}(1.731^nn^{{\mathcal {O}}(1)})\).