Algorithmic Universality, Low-Degree Polynomials, and Max-Cut in Sparse Random Graphs

TL;DR

This paper proves that certain low-degree polynomial algorithms exhibit universal performance across different random graph models, specifically for Max-Cut in sparse Erdős-Rényi graphs, by establishing their asymptotic behavior matches that in mean-field models.

Contribution

It introduces the concept of algorithmic universality for low-degree polynomial algorithms and demonstrates this for Max-Cut in sparse graphs, linking performance in Erdős-Rényi graphs to the SK model.

Findings

LDP algorithms have universal performance in different models.

Performance of LDP algorithms is consistent between SK and sparse graphs.

Discreteness of LDP outputs is established via universality of coordinate-wise statistics.

Abstract

Universality, namely distributional invariance, is a well-known property for many random structures. For example, it is known to hold for a broad range of variational problems with random input. Much less is known about the algorithmic universality of specific methods for solving such variational problems. Namely, whether algorithms tuned to specific variational tasks produce the same asymptotic behavior across different input distributions with matching moments. In this paper, we establish algorithmic universality for a class of models, which includes spin glass models and constraint satisfaction problems on sparse graphs, provided that an algorithm can be coded as a low-degree polynomial (LDP). We illustrate this specifically for the case of the Max-Cut problem in sparse Erd\"os-R\'enyi graph $\mathbb{G}(n,d/n)$. We use the fact that the Approximate Message Passing (AMP) algorithm, which is an effective algorithm for finding near-ground states of the Sherrington-Kirkpatrick (SK) model, is well approximated by an LDP. We then establish our main universality result: the performance of the LDP based algorithms exhibiting a certain connectivity property, is the same in the mean-field (SK) and in the random graph $\mathbb{G}(n,d/n)$ setting, up to an appropriate rescaling. The main technical challenge we address in this paper is showing that the output of an LDP algorithm on $\mathbb{G}(n,d/n)$ is truly discrete, namely, that it is close to the set of points in the binary cube. This is achieved by establishing universality of coordinate-wise statistics of the LDP output across disorder ensembles, which implies that proximity to the cube transfers from the Gaussian to the sparse graph setting.