Ruling Sets in Random Order and Adversarial Streams

Abstract

The goal of this paper is to understand the complexity of a key symmetry breaking problem, namely the ( α, β )-ruling set problem in the graph streaming model. Given a graph G = ( V, E ), an ( α, β )-ruling set is a subset I ⊆ V such that the distance between any two vertices in I is at least α and the distance between a vertex in V and the closest vertex in I is at most β . This is a fundamental problem in distributed computing where it finds applications as a useful subroutine for other problems such as maximal matching, distributed colouring, or shortest paths. Additionally, it is a generalization of MIS, which is a (2 , 1)-ruling set. Our main results are two algorithms for (2 , 2)-ruling sets: adversarial order in which edges arrive arbitrary, we give an n 3 upon the best known algorithm to Konrad et al. [DISC 2019], Finally, we present new algorithms and lower bounds for ( α, β )-ruling sets for other values of α and β . Our algorithms improve and generalize the previous work of Konrad et al. [DISC 2019] for (2 , β )-ruling sets, while our lower bound establishes the impossibility of obtaining any non-trivial streaming algorithm for ( α, α − 1)-ruling sets for all even α > 2.