Parameterized Complexity of Red Blue Set Cover for Lines

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, when parameterized by \(k_\ell +k_r\). In this paper, we study Gen-RBSC-lines, where the elements are points in the plane and sets are defined by lines. We study this problem for the parameters \(k_\ell , k_r\), and \(k_\ell +k_r\). For all these cases, we either prove that the problem is W-hard or show that the problem is fixed parameter tractable (FPT). Finally, for the parameter \(k_\ell +k_r\), for which Gen-RBSC-lines admits FPT algorithms, we show that the problem does not have a polynomial kernel unless \(\text { co-NP}\subseteq \text { NP}/\mathrm{poly}\). Further, we show that the FPT algorithm does not generalize to higher dimensions.