Multivariate Complexity Analysis of Geometric Red Blue Set Cover

We investigate the parameterized complexity of Generalized Red Blue Set Cover (Gen-RBSC), a generalization of the classic Set Cover problem and the more recently studied Red Blue Set Cover problem. Given a universe U containing b blue elements and r red elements, positive integers k_ℓ and k_r, and a family F of ℓ sets over U, the Gen-RBSC problem is to decide whether there is a subfamily F^′⊆ F of size at most k_ℓ that covers all blue elements, but at most k_r of the red elements. This generalizes Set Cover and thus in full generality it is intractable in the parameterized setting. In this paper, we study a geometric version of this problem, called Gen-RBSC-lines, where the elements are points in the plane and sets are defined by lines. We study this problem for an array of parameters, namely, k_ℓ, k_r, r, b, and ℓ, and all possible combinations of them. For all these cases, we either prove that the problem is W-hard or show that the problem is fixed parameter tractable (FPT). In particular, on the algorithmic side, our study shows that a combination of k_ℓ and k_r gives rise to a nontrivial algorithm for Gen-RBSC-lines. On the hardness side, we show that the problem is para-NP-hard when parameterized by k_r, and W[1]-hard when parameterized by k_ℓ. Finally, for the combination of parameters for which Gen-RBSC-lines admits FPT algorithms, we ask for the existence of polynomial kernels. We are able to provide a complete kernelization dichotomy by either showing that the problem admits a polynomial kernel or that it does not contain a polynomial kernel unless co-NP⊆NP/poly.