Algorithmic Universality, Low-Degree Polynomials, and Max-Cut in Sparse Random Graphs
Houssam El Cheairi, David Gamarnik
TL;DR
The paper establishes algorithmic universality for a broad class of Low-Degree Polynomial (LDP) based algorithms solving variational problems with random input, focusing on Max-Cut in sparse graphs. By showing that AMP-type procedures can be well approximated by connected LDPs and transferring state-evolution predictions from Gaussian to sparse ensembles via Lindeberg interpolation, the authors prove that LDP-based performance is invariant under input distributions with matched first two moments. They formalize this through two main results: (i) a universality principle for LDP outputs across GOE and sparse random graphs, and (ii) a rounding scheme that yields near-optimal Max-Cut values on sparse graphs. A key technical contribution is establishing universality of coordinate-wise statistics of LDP outputs, which enables the rounding argument and connects discrete optimization on sparse graphs to mean-field spin-glass theory. The work advances understanding of when algorithmic performance is input-distribution invariant and provides a principled path from AMP to practical LDP rounding in sparse settings.
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 Erdös-Rényi 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.