Approximability of covering cells with line segments

Paz Carmi, Anil Maheshwari, Saeed Mehrabi, Luís Fernando Schultz Xavier da Silveira

Research output: Contribution to journalArticlepeer-review

Abstract

Korman et al. [18] 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 to 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. [18] in which the goal is to cover the “rectangular” cells.

Original languageEnglish
Pages (from-to)133-141
Number of pages9
JournalTheoretical Computer Science
Volume784
DOIs
StatePublished - 13 Sep 2019

Keywords

  • APX-hardness
  • Approximation algorithms
  • Fixed-parameter tractability
  • Geometric covering
  • Line segment covering
  • PTAS

Fingerprint

Dive into the research topics of 'Approximability of covering cells with line segments'. Together they form a unique fingerprint.

Cite this