Leveraging Linial’s Locality Limit

Abstract

In this paper we extend the lower bound technique by Linial for local coloring and maximal independent sets. We show that constant approximations to maximum independent sets on a ring require at least log-star time. More generally, the product of approximation quality and running time cannot be less than log-star. Using a generalized ring topology, we gain identical lower bounds for approximations to minimum dominating sets. Since our generalized ring topology is contained in a number of geometric graphs such as the unit disk graph, our bounds directly apply as lower bounds for quite a few algorithmic problems in wireless networking.

Having in mind these and other results about local approximations of maximum independent sets and minimum dominating sets, one might think that the former are always at least as difficult to obtain as the latter. Conversely, we show that graphs exist, where a maximum independent set can be determined without any communication, while finding even an approximation to a minimum dominating set is as hard as in general graphs.