Line Segment Covering of Cells in Arrangements

Abstract

Given a collection L of line segments, we consider its arrangement and study the problem of covering all cells with line segments of L. That is, we want to find a minimum-size set L of line segments such that every cell in the arrangement has a line from L defining its boundary. We show that the problem is NP-hard, even when all segments are axis-aligned. In fact, the problem is still NP-hard when we only need to cover rectangular cells of the arrangement. For the latter problem we also show that it is fixed parameter tractable with respect to the size of the optimal solution. Finally we provide a linear time algorithm for the case where cells of the arrangement are created by recursively subdividing a rectangle using horizontal and vertical cutting segments. * A preliminary version of this paper appeared in the proceedings of the COCOA 2015 conference [8]

Approximability of Covering Cells with Line Segments

“…In this section, we show that the LSC problem is fixed-parameter tractable (parametrized by the size of an optimal solution) when the line segments in S are either horizontal or vertical, and the goal is to cover all the cells in A(S). This complements the FPT result of Korman et al [9], where the goal is to cover the rectangular cells. Throughout this section, let k be the size of an optimal solution.…”

“…• We give an FPT algorithm for the LSC problem when the line segments in S have only two orientations and the goal is to cover all cells of the arrangement. This complements the FPT algorithm of Korman et al [9] as we do not restrict the covering only to rectangular cells.…”

“…In this paper, we consider a geometric variant of the set cover problem that was first studied by Korman et al [9]. A set of line segments in the plane is said to be non-overlapping if any two line segments from the set intersect in at most one point.…”

“…• We give a PTAS for the LSC problem when the line segments in S can have any arbitrarily orientations, but we are allowed to select the covering line segments from only one orientation. Given the NP-hardness of the problem [9], this settles the complexity of this variant of the problem.…”

“…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 [9] in which the goal is to cover the "rectangular" cells.…”