Finding Dense Subgraphs in G(n,1/2)

TL;DR

This paper investigates the problem of finding dense subgraphs in random graphs, demonstrating that a modified greedy algorithm can find relatively large dense subgraphs, while larger ones are provably absent.

Contribution

It introduces a simple modification to the greedy algorithm that efficiently finds large dense subgraphs in G(n,1/2), and establishes bounds on the maximum size of such subgraphs.

Findings

Modified greedy algorithm finds dense subgraphs of size 2log n

No dense subgraph of size 2.784log n exists with high probability

Dense subgraphs of size 2log n have edge density at least 0.951

Abstract

Finding the largest clique is a notoriously hard problem, even on random graphs. It is known that the clique number of a random graph G(n,1/2) is almost surely either k or k+1, where k = 2log n - 2log(log n) - 1. However, a simple greedy algorithm finds a clique of size only (1+o(1))log n, with high probability, and finding larger cliques -- that of size even (1+ epsilon)log n -- in randomized polynomial time has been a long-standing open problem. In this paper, we study the following generalization: given a random graph G(n,1/2), find the largest subgraph with edge density at least (1-delta). We show that a simple modification of the greedy algorithm finds a subset of 2log n vertices whose induced subgraph has edge density at least 0.951, with high probability. To complement this, we show that almost surely there is no subset of 2.784log n vertices whose induced subgraph has edge density 0.951 or more.