Phase transition in the maximum clique problem: the case of Erdos-Renyi graphs

TL;DR

This paper investigates a phase transition phenomenon in the maximum clique problem on Erdős-Rényi graphs, revealing finite size scaling behavior and instances requiring extensive computational effort.

Contribution

It identifies and analyzes a phase transition in the max clique problem on ER graphs, highlighting its finite size scaling and computational complexity implications.

Findings

A phase transition exists in the max clique problem on ER graphs.

The transition exhibits finite size scaling behavior.

Certain graph instances require significantly more CPU time to solve.

Abstract

A phase transition, like the one already found on Boolean satisfiability problem by Kirkpatrick and Selman, is found on max clique problem on ER graphs. Although number of the datapoints is limited, the transition seems to obey finite size scaling. The transition also shows concentration of the graph instances which need particularly large CPU time to solve.