Parameterized Complexity of d-Hitting Set with Quotas

In this paper we study a variant of the classic d -Hitting Set problem with lower and upper capacity constraints, say A and B, respectively. The input to the problem consists of a universe U, a set family, \(\mathscr {S} \), of sets over U, where each set in the family is of size at most d, a non-negative integer k; and additionally two functions \(\alpha :\mathscr {S} \rightarrow \{1,\ldots ,A\}\) and \(\beta :\mathscr {S} \rightarrow \{1,\ldots ,B\}\). The goal is to decide if there exists a hitting set of size at most k such that for every set S in the family \(\mathscr {S} \), the solution contains at least \(\alpha (S)\) elements and at most \(\beta (S)\) elements from S. We call this the \((A, B)\)-Multi d-Hitting Set problem. We study the problem in the realm of parameterized complexity. We show that \((A, B)\)-Multi d-Hitting Set can be solved in \(\mathcal {O}^{\star }(d^{k}) \) time. For the special case when \(d=3\) and \(d=4\), we have an improved bound of \(\mathcal {O}^\star (2.2738^k)\) and \(\mathcal {O}^\star (3.562^{k})\), respectively. The former matches the running time of the classical 3-Hitting Set problem. Furthermore, we show that if we do not have an upper bound constraint and the lower bound constraint is same for all the sets in the family, say \(A>1\), then the problem can be solved even faster than d-Hitting Set.