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

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.