Multivariate Complexity Analysis of Geometric Red Blue Set Cover

Abstract

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 \(k_\ell \) and \(k_r\), and a family \(\mathcal F \) of \(\ell \) sets over U, the Gen-RBSC problem is to decide whether there is a subfamily \(\mathcal F '\subseteq \mathcal F \) of size at most \(k_\ell \) 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, \(k_\ell , k_r, r, b\), and \(\ell \), 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 \(k_\ell \) 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 \(k_\ell \). 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 \(\text {co{-}NP}\subseteq \text {NP}/\text{ poly }\).