Sequential Algorithms and Independent Sets Discovering on Large Sparse Random Graphs

TL;DR

This paper develops and analyzes sequential, degree-aware algorithms for estimating the maximum independent set size in large sparse random graphs, providing hydrodynamic limits and applications to wireless network capacity.

Contribution

It introduces static and dynamic degree-aware exploration algorithms with hydrodynamic limits, enabling efficient approximation of independence numbers in sparse random graphs.

Findings

Hydrodynamic limits derived for both algorithms.

Algorithms can be implemented in a distributed manner.

Method applied to estimate wireless network capacity.

Abstract

Computing the size of maximum independent sets is a NP-hard problem for fixed graphs. Characterizing and designing efficient algorithms to estimate this independence number for random graphs are notoriously difficult and still largely open issues. In a companion paper, we showed that a low complexity degree-greedy exploration is actually asymptotically optimal on a large class of sparse random graphs. Encouraged by this result, we present and study two variants of sequential exploration algorithms: static and dynamic degree-aware explorations. We derive hydrodynamic limits for both of them, which in turn allow us to compute the size of the resulting independent set. Whereas the former is simpler to compute, the latter may be used to arbitrarily approximate the degree-greedy algorithm. Both can be implemented in a distributed manner. The corresponding hydrodynamic limits constitute an efficient method to compute or bound the independence number for a large class of sparse random graphs. As an application, we then show how our method may be used to estimate the capacity of a large 802.11-based wireless network. We finally consider further indicators such as the fairness of the resulting configuration, and show how an unexpected trade-off between fairness and capacity can be achieved.