Constrained hitting set problem with intervals: Hardness, FPT and approximation algorithms

We study a constrained version of the GEOMETRIC HITTING SET problem where we are given a set of points, partitioned into pairwise disjoint subsets, and a set of intervals. The objective is to hit all the intervals with a minimum number of points such that if we select a point from a subset, we must select all the points from that subset. We consider two special cases of the problem where each subset can have at most 2 and 3 points. If each subset contains at most 2 points and the intervals are disjoint, we show that the problem admits a polynomial -time algorithm. On the contrary, if each subset contains at most t points, where t >= 2, and the intervals are overlapping, we show that the problem becomes NP -Hard. Further, when each subset contains at most t points where t >= 3, and the intervals are disjoint, we prove that the problem is NP -Hard, and we provide two constant factor approximation algorithms for this problem. We also study the problem from the parameterized complexity perspective. If the intervals are disjoint, then we prove that the problem is in FPT when parameterized by the size of the solution. We also complement this result by giving a lower bound in the size of the kernel for disjoint intervals, and we also provide a polynomial kernel when the size of all subsets is bounded by a constant.