The algorithmic hardness threshold for continuous random energy models

Abstract

We prove an algorithmic hardness result for finding low-energy states in the so-called continuous random energy model (CREM) , introduced by Bovier and Kurkova in 2004 as an extension of Derrida’s generalized random energy model . The CREM is a model of a randomenergy landscape (X_v)_{v \in \{0,1\}^N} on the discrete hypercube with built-in hierarchical structure, and can be regarded as a toy model for strongly correlated random energy landscapes such as the family of p -spin models including the Sherrington–Kirkpatrick model. The CREM is parameterized by an increasing function A \colon [0,1]\to[0,1] , which encodes the correlations between states. We exhibit an algorithmic hardness threshold x_* , which is explicit in terms of A . More precisely, we obtain two results: First, we show that a renormalization procedure combined with a greedy search yields for any \varepsilon > 0 a linear-time algorithm which finds states v \in \{0,1\}^N with X_v \ge (x_*-\varepsilon) N . Second, we show that the value x_* is essentially best-possible: for any \varepsilon > 0 , any algorithm which finds states v with X_v \ge (x_*+\varepsilon)N requires exponentially many queries in expectation and with high probability. We further discuss what insights this study yields for understanding algorithmic hardness thresholds for random instances of combinatorial optimization problems.