Parameterized Complexity of Geometric Covering Problems Having Conflicts

Abstract

The input for the Geometric Coverage problem consists of a pair $\varSigma =(P,\mathcal {R})$, where P is a set of points in $\mathbb {R}^d$ and $\mathcal {R}$ is a set of subsets of P defined by the intersection of P with some geometric objects in $\mathbb {R}^d$. These coverage problems form special instances of the Set Cover problem which is notoriously hard in several paradigms including approximation and parameterized complexity. Motivated by what are called choice problems in geometry, we consider a variation of the Geometric Coverage problem where there are conflicts on the covering objects that precludes some objects from being part of the solution if some others are in the solution.

As our first contribution, we propose two natural models in which the conflict relations are given: (a) by a graph on the covering objects, and (b) by a representable matroid on the covering objects. We consider the parameterized complexity of the problem based on the structure of the conflict relation. Our main result is that as long as the conflict graph has bounded arboricity (that includes all the families of intersection graphs of low density objects in low dimensional Euclidean space), there is a parameterized reduction to the problem without conflicts on the covering objects. This is achieved through a randomization-derandomization trick. As a consequence, we have the following results when the conflict graph has bounded arboricity.

• If the Geometric Coverage problem is fixed parameter tractable (FPT), then so is the conflict free version.

• If the Geometric Coverage problem admits a factor $\alpha$-approximation, then the conflict free version admits a factor $\alpha$-approximation algorithm running in FPT time.

As a corollary to our main result we get a plethora of approximation algorithms running in FPT time. Our other results include an FPT algorithm and a W[1]-hardness proof for the conflict-free version of Covering Points by Intervals. The FPT algorithm is for the case when the conflicts are given by a representable matroid, and the W[1]-hardness result is for all the families of conflict graphs for which the Independent Set problem is W[1]-hard.

Supported by Parameterized Approximation, ERC Starting Grant 306992, and Rigorous Theory of Preprocessing, ERC Advanced Investigator Grant 267959.