The Peculiar Phase Structure of Random Graph Bisection

TL;DR

This paper investigates the phase transition in the mincut graph bisection problem on sparse random graphs, revealing new bounds and suggesting the problem remains replica symmetric beyond the critical threshold, with implications for polynomial-time approximation.

Contribution

It introduces a new analytical upper bound for the minimum cutsize and explores the phase structure, including potential replica symmetry breaking, in the graph bisection problem.

Findings

New upper bound on cutsize above critical mean degree

Evidence of replica symmetry beyond the critical threshold

Polynomial-time near-optimal solutions near phase transition

Abstract

The mincut graph bisection problem involves partitioning the n vertices of a graph into disjoint subsets, each containing exactly n/2 vertices, while minimizing the number of "cut" edges with an endpoint in each subset. When considered over sparse random graphs, the phase structure of the graph bisection problem displays certain familiar properties, but also some surprises. It is known that when the mean degree is below the critical value of 2 log 2, the cutsize is zero with high probability. We study how the minimum cutsize increases with mean degree above this critical threshold, finding a new analytical upper bound that improves considerably upon previous bounds. Combined with recent results on expander graphs, our bound suggests the unusual scenario that random graph bisection is replica symmetric up to and beyond the critical threshold, with a replica symmetry breaking transition possibly taking place above the threshold. An intriguing algorithmic consequence is that although the problem is NP-hard, we can find near-optimal cutsizes (whose ratio to the optimal value approaches 1 asymptotically) in polynomial time for typical instances near the phase transition.