Breaking of 1RSB in Random Regular MAX-NAE-SAT

Abstract

A. For several models of random constraint satisfaction problems, it was conjectured by physicists and later proved that a sharp satis ability transition occurs. For random k-and related models it happens at clause density α α sat 2 k . Just below the threshold, further results suggest that the solution space has a " " structure of a large bounded number of near-orthogonal clusters inside {0, 1} N .In the unsatis able regime α > α sat , it is natural to consider the problem of max-satis ability: violating the least number of constraints.is is a combinatorial optimization problem on the random energy landscape de ned by the problem instance. For a simpli ed variant, the strong refutation problem, there is strong evidence that an algorithmic transition occurs around α N k/2−1 . For α bounded in N, a very precise estimate of the max-sat value was obtained by Achlioptas, Naor, and Peres (2007), but it is not sharp enough to indicate the nature of the energy landscape. Later work (Sen, 2016; Panchenko, 2016) shows that for α very large (roughly, Ω(64 k )) the max-sat value approaches the mean-eld (complete graph) limit: this is conjectured to have an " " structure where near-optimal con gurations form clusters within clusters, in an ultrametric hierarchy of in nite depth inside {0, 1} N . A stronger form of was shown in several recent works to have algorithmic implications (again, in complete graphs). Consequently we nd it of interest to understand how the model transitions from near α sat , to (conjecturally) for large α. In this paper we show that in the random regular kmodel, the description breaks down already above α 4 k /k 3 . is is proved by an explicit perturbation in the parameter space. e choice of perturbation is inspired by the "bug proliferation" mechanism proposed by physicists (Montanari and Ricci-Tersenghi, 2003;Krzakala, Pagnani, and Weigt, 2004), corresponding roughly to a percolation-like threshold for a subgraph of dependent variables.

The Ising antiferromagnet and max cut on random regular graphs

“…Yet this result says nothing about any finite d and, indeed, sparse random graphs for fixed finite values of d appear to exhibit a more diverse and potentially even more intricate behavior. As a result, we are only just beginning to understand the genuine behaviour of sparse models in the replica symmetry breaking phase; see, e.g., [6]. Finally, Panchenko [54] obtained a variational formula for the free energy of the Ising antiferromagnet on Erdős-Rényi graphs.…”

Frozen $1$-RSB structure of the symmetric Ising perceptron

“…Studying symmetric constraints has a long history in the random CSP literature: the random k-NAE-SAT model is a symmetric variant of the random k-SAT model. While the qualitative properties of the two models are expected to be very similar, the symmetric model is often more amenable to rigorous analysis, and thus a clearer understanding can be obtained (see e.g., [DSS14,SSZ16,BSZ19] for recent work on the k-NAE-SAT model). Studying symmetric perceptrons allows us to prove stronger and more detailed results than are currently attainable for the classic perceptron, but the phenomena studied are expected to be universal.…”

The Ising Antiferromagnet and Max Cut on Random Regular Graphs

“…Yet this result says nothing about any finite d and, indeed, sparse random graphs for fixed finite values of d appear to exhibit a more diverse and potentially even more intricate behavior. As a result, we are only just beginning to understand the genuine behaviour of sparse models in the replica symmetry breaking phase; see, e.g., [7]. Finally, Panchenko [58] obtained a variational formula for the free energy of the Ising antiferromagnet on Erdős-Rényi graphs.…”