Extremal Optimization at the Phase Transition of the 3-Coloring Problem

TL;DR

This paper studies the phase transition in the 3-coloring problem on random graphs using extremal optimization, revealing a first-order transition and providing insights into the problem's complexity at the critical point.

Contribution

It applies extremal optimization to analyze the 3-coloring phase transition, estimating the critical point and characterizing the nature of the transition.

Findings

Critical mean degree $oldsymbol{ ext{α}_c=4.703(28)}$ identified.

Extremal optimization effectively finds ground states near the transition.

The backbone order parameter exhibits a first-order phase transition.

Abstract

We investigate the phase transition of the 3-coloring problem on random graphs, using the extremal optimization heuristic. 3-coloring is among the hardest combinatorial optimization problems and is closely related to a 3-state anti-ferromagnetic Potts model. Like many other such optimization problems, it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter: the graph's mean vertex degree. This phase transition is often associated with the instances of highest complexity. We use extremal optimization to measure the ground state cost and the ``backbone'', an order parameter related to ground state overlap, averaged over a large number of instances near the transition for random graphs of size $n$ up to 512. For graphs up to this size, benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about $O(n^{3.5})$ update steps. Finite size scaling gives a critical mean degree value $\alpha_{\rm c}=4.703(28)$. Furthermore, the exploration of the degenerate ground states indicates that the backbone order parameter, measuring the constrainedness of the problem, exhibits a first-order phase transition.