Distributed Independent Sets in Interval and Segment Intersection Graphs

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 inO(k) rounds andO(n) messages, wherek is the size of the maximum independent set andn is the number of nodes in the graph. We provide a matching lower bound ofω(k) on the number of rounds, whereasω(n) 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 countl graphs, a special case of the interval graphs where the intervals have exactlyl different lengths. We propose an1 2-approximation algorithm that runs inO(l) round. For axis-parallel segment intersection graphs, we design an1 2-approximation algorithm that obtains a solution inO(D) rounds. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].