On approximating partial scenario set cover

The Partial Scenario Set Cover problem (PSSC) generalizes the Partial Set Cover problem, which is itself a generalization of the classical Set Cover problem. We are given a finite ground set Q, a collection $S$ of subsets of Q to choose from, each of which is associated with a nonnegative cost, and a second collection $U$ of subsets of Q of which a given number l must be covered. The task is to choose a minimum cost sub-collection from $S$ that covers at least l sets from $U$. PSSC is motivated by an application for locating emergency doctors.

We present two approximation approaches. The first one combines LP-based rounding with a greedy consideration of the scenarios. The other is a variant of the greedy set cover algorithm, and in each iteration tries to minimize the ratio of cost to number of newly covered scenarios. We show that this subproblem, which we call Dense Scenario Set Cover (DSSC), is itself as hard to approximate as Set Cover and NP-hard, even when there is only a single scenario and all sets contain at most three elements. Furthermore, we consider a special case of DSSC where the sets are pairwise disjoint and show that in this case DSSC can be solved in polynomial time. We also provide an approximation for the general case, which we use as a subroutine in the greedy algorithm to obtain an approximation for PSSC.