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.

S. Saurabh—The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 306992.