Aurélien Decelle, Janina Hüttel, Alaa Saade, Cristopher Moore
Published 2014-09-08Version 1
We take a whirlwind tour of problems and techniques at the boundary of computer science and statistical physics. We start with a brief description of P, NP, and NP-completeness. We then discuss random graphs, including the emergence of the giant component and the k-core, using techniques from branching processes and differential equations. Using these tools as well as the second moment method, we give upper and lower bounds on the critical clause density for random k-SAT. We end with community detection in networks, variational methods, the Bethe free energy, belief propagation, the detectability transition, and the non-backtracking matrix.
Comments: Chapter of "Statistical Physics, Optimization, Inference, and Message-Passing Algorithms", Eds.: F. Krzakala, F. Ricci-Tersenghi, L. Zdeborova, R. Zecchina, E. W. Tramel, L. F. Cugliandolo (Oxford University Press, to appear)
Classification on the Computational Complexity of Spin Models
Phase Transitions of Traveling Salesperson Problems solved with Linear Programming and Cutting Planes
$K$-selective percolation: A simple model leading to a rich repertoire of phase transitions
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