Distributed Independent Sets in Interval and Segment Intersection Graphs

Abstract

The Maximum Independent Set problem is well-studied in graph theory and related areas. An independent set of a graph is a subset of non-adjacent vertices of the graph. A maximum independent set is an independent set of maximum size. This paper studies the Maximum Independent Set problem in some classes of geometric intersection graphs in a distributed setting. More precisely, we study the Maximum Independent Set problem on two geometric intersection graphs, interval and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in a model that is similar but a little weaker than the local communication model. We compute the maximum independent set on interval graphs in [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] is the size of the maximum independent set and [Formula: see text] is the number of nodes in the graph. We provide a matching lower bound of [Formula: see text] on the number of rounds, whereas [Formula: see text] is a trivial lower bound on message complexity. Thus, our algorithm is both time and message-optimal. We also study the Maximum Independent Set problem in interval count [Formula: see text] graphs, a special case of the interval graphs where the intervals have exactly [Formula: see text] different lengths. We propose an [Formula: see text]-approximation algorithm that runs in [Formula: see text] round. For axis-parallel segment intersection graphs, we design an [Formula: see text]-approximation algorithm that obtains a solution in [Formula: see text] rounds. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].