On independent sets in random graphs

TL;DR

This paper investigates the structure of independent sets in sparse random graphs, revealing a phase transition that explains why certain algorithms struggle to find larger independent sets beyond a threshold.

Contribution

It proves a phase transition in the structure of independent sets in random graphs and shows local search algorithms are inherently limited beyond a certain size.

Findings

Independent sets of size above a threshold form a complex landscape.

Local search algorithms are likely to get stuck when finding larger independent sets.

Exponential lower bounds are established for the Metropolis process.

Abstract

The independence number of a sparse random graph G(n,m) of average degree d=2m/n is well-known to be \alpha(G(n,m))~2n ln(d)/d with high probability. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size (1+o(1)) n ln(d)/d, i.e. half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with (1+c)n ln(d)/d, for any fixed c>0. In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size k of the independent sets passes the point k nln(d)/d. Roughly speaking, we prove that independent sets of size k>(1+c)n ln(d)/d form an intricately ragged landscape, in which local search algorithms are bound to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independents sets.