Algorithms for Intersection Graphs of Multiple Intervals and Pseudo Disks

TL;DR

This paper improves approximation algorithms for minimum weight dominating sets and maximum weight independent sets in intersection graphs of multiple intervals and pseudo-disks, extending results to broader geometric classes.

Contribution

It introduces tighter approximation bounds for these problems on multiple-interval and pseudo-disk intersection graphs, surpassing previous bounds and establishing NP-hardness results.

Findings

O(t log t) approximation for minimum weight dominating set

NP-hardness of o(t)-approximation for the same problem

Omega(1/t) approximation for maximum weight independent set

Abstract

Intersection graphs of planar geometric objects such as intervals, disks, rectangles and pseudo-disks are well studied. Motivated by various applications, Butman et al. in SODA 2007 considered algorithmic questions in intersection graphs of $t$-intervals. A $t$-interval is a union of at most $t$ distinct intervals (here $t$ is a parameter) -- these graphs are referred to as Multiple-Interval Graphs. Subsequent work by Kammer et al. in Approx 2010 also considered $t$-disks and other geometric shapes. In this paper we revisit some of these algorithmic questions via more recent developments in computational geometry. For the minimum weight dominating set problem, we give a simple $O(t \log t)$ approximation for Multiple-Interval Graphs, improving on the previously known bound of $t^2$ . We also show that it is NP-hard to obtain an $o(t)$-approximation in this case. In fact, our results hold for the intersection graph of a set of t-pseudo-disks which is a much larger class. We obtain an ${\Omega}(1/t)$-approximation for the maximum weight independent set in the intersection graph of $t$-pseudo-disks. Our results are based on simple reductions to existing algorithms by appropriately bounding the union complexity of the objects under consideration.