Hard Optimization Problems have Soft Edges

TL;DR

This paper investigates the structure of the Maximum Clique problem in Erdős-Rényi graphs, revealing a complex phase boundary and demonstrating that local algorithms can find larger cliques than previously expected, especially in finite systems.

Contribution

It uncovers the detailed phase boundary structure of the Maximum Clique problem and shows local algorithms can outperform spectral methods in finite-size graphs.

Findings

Complex staircase phase boundary in clique size versus graph parameters

Finite-width boundaries enable local algorithms to find larger cliques

Early stopping local searches outperform spectral algorithms in hidden clique detection

Abstract

Finding a Maximum Clique is a classic property test from graph theory; find any one of the largest complete subgraphs in an Erd\"os-R\'enyi G(N, p) random graph. We use Maximum Clique to explore the structure of the problem as a function of N, the graph size, and K, the clique size sought. It displays a complex phase boundary, a staircase of steps at each of which 2log2 N and Kmax, the maximum size of a clique that can be found, increases by 1. Each of its boundaries has a finite width, and these widths allow local algorithms to find cliques beyond the limits defined by the study of infinite systems. We explore the performance of a number of extensions of traditional fast local algorithms, and find that much of the "hard" space remains accessible at finite N. The "hidden clique" problem embeds a clique somewhat larger than those which occur naturally in a G(N, p) random graph. Since such a clique is unique, we find that local searches which stop early, once evidence for the hidden clique is found, may outperform the best message passing or spectral algorithms.