The algorithmic hardness threshold for continuous random energy models
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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 random energy landscape (X_v)_v ∈{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:[0,1]→[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 ε > 0 a linear-time algorithm which finds states v ∈{0,1}^N with X_v > (x_*-ε) N. Second, we show that the value x_* is essentially best-possible: for any ε > 0, any algorithm which finds states v with X_v > (x_*+ε)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.
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