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Volume 18 (2022) Article 18 pp. 1-19
Algorithms for Intersection Graphs for t-Intervals and t-Pseudodisks
Received: December 19, 2019
Revised: June 8, 2022
Published: June 23, 2022
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Keywords: approximation algorithms, multiple interval graphs, geometric set cover, dominating set, independent set
ACM Classification: F.5.2.2, F.6.2
AMS Classification: 68W25

Abstract: [Plain Text Version]

\newcommand{\NP}{\mathsf{NP}}

Intersection graphs of planar geometric objects such as intervals, disks, rectangles and pseudodisks are well-studied. Motivated by various applications, Butman et al. (ACM Trans. Algorithms, 2010) considered algorithmic questions in intersection graphs of t-intervals. A t-interval is a union of t intervals—these graphs are also referred to as multiple-interval graphs. Subsequent work by Kammer et al. (APPROX-RANDOM 2010) considered intersection graphs of t-disks (union of t disks), and other geometric objects. In this paper we revisit some of these algorithmic questions via more recent developments in computational geometry. For the minimum-weight dominating set problem in t-interval graphs, we obtain a polynomial-time O(t \log t)-approximation algorithm, improving upon the previously known polynomial-time t^2-approximation by Butman et al. (op. cit.). In the same class of graphs we show that it is \NP-hard to obtain a (t-1-\epsilon)-approximation for any fixed t \ge 3 and \epsilon > 0.

The approximation ratio for dominating set extends to the intersection graphs of a collection of t-pseudodisks (nicely intersecting t-tuples of closed Jordan domains). We obtain an \Omega(1/t)-approximation for the maximum-weight independent set in the intersection graph of t-pseudodisks in polynomial time. Our results are obtained via simple reductions to existing algorithms by appropriately bounding the union complexity of the objects under consideration.