Suboptimality of local algorithms for a class of max-cut problems

TL;DR

This paper demonstrates that local algorithms cannot find nearly maximal cuts in certain random hypergraphs with large average degree, due to an overlap gap property that limits their effectiveness.

Contribution

It establishes the suboptimality of local algorithms for max-cut in random hypergraphs by proving an overlap gap property through spin glass model comparisons.

Findings

Local algorithms fail to find nearly maximal cuts in large-degree hypergraphs.

Overlap gap property prevents local algorithms from achieving optimal cuts.

Comparison with spin glass models confirms the overlap gap phenomenon.

Abstract

We show that in random $K$-uniform hypergraphs of constant average degree, for even $K \geq 4$, local algorithms defined as factors of i.i.d. can not find nearly maximal cuts, when the average degree is sufficiently large. These algorithms have been used frequently to obtain lower bounds for the max-cut problem on random graphs, but it was not known whether they could be successful in finding nearly maximal cuts. This result follows from the fact that the overlap of any two nearly maximal cuts in such hypergraphs does not take values in a certain non-trivial interval - a phenomenon referred to as the overlap gap property - which is proved by comparing diluted models with large average degree with appropriate fully connected spin glass models and showing the overlap gap property in the latter setting.