Hitting and Covering Partially

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 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 subset of U of size at most k such that S intersects with at least t sets in F. We obtain a kernel for PARTIAL d-HS in which the size of the universe is bounded by O(dt) and the size of the family is bounded by O(dt(2)). Using this result, we obtain a kernel for PARTIAL VERTEX COVER (PVC) with 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 F, of sets of size exactly d over U, and integers k and t, the objective is to decide if there is S subset of F of size at most k, such that 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 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 O(1.731(n)n(O(1))).