Approximability of Covering Cells with Line Segments

TL;DR

This paper studies the computational complexity of covering cells in a line segment arrangement with minimal segments, providing approximation schemes, hardness results, and fixed-parameter tractability insights.

Contribution

It introduces a PTAS for covering cells with segments of any orientation from one direction, proves APX-hardness for covering rectangular cells with both orientations, and offers an FPT algorithm for certain cases.

Findings

PTAS developed for segments in any orientation from one direction

Proved APX-hardness for covering rectangular cells with both orientations

Designed an FPT algorithm for covering all cells with two orientations

Abstract

In COCOA 2015, Korman et al. studied the following geometric covering problem: given a set $S$ of $n$ line segments in the plane, find a minimum number of line segments such that every cell in the arrangement of the line segments is covered. Here, a line segment $s$ covers a cell $f$ if $s$ is incident to $f$. The problem was shown to be NP-hard, even if the line segments in $S$ are axis-parallel, and it remains NP-hard when the goal is cover the "rectangular" cells (i.e., cells that are defined by exactly four axis-parallel line segments). In this paper, we consider the approximability of the problem. We first give a PTAS for the problem when the line segments in $S$ are in any orientation, but we can only select the covering line segments from one orientation. Then, we show that when the goal is to cover the rectangular cells using line segments from both horizontal and vertical line segments, then the problem is APX-hard. We also consider the parameterized complexity of the problem and prove that the problem is FPT when parameterized by the size of an optimal solution. Our FPT algorithm works when the line segments in $S$ have two orientations and the goal is to cover all cells, complementing that of Korman et al. in which the goal is to cover the "rectangular" cells.