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Exact solutions for diluted spin glasses and optimization problems

S. Franz·M. Leone·F. Ricci-Tersenghi·R. Zecchina
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Abstract

We study the low temperature properties of p-spin glass models with finite connectivity and of some optimization problems. Using a one-step functional replica symmetry breaking Ansatz we can solve exactly the saddle-point equations for graphs with uniform connectivity. The resulting ground state energy is in perfect agreement with numerical simulations. For fluctuating connectivity graphs, the same Ansatz can be used in a variational way: For p-spin models (known as p-XOR-SAT in computer science) it provides the exact configurational entropy together with the dynamical and static critical connectivities (for p=3, γ_d=0.818 and γ_s=0.918 resp.), whereas for hard optimization problems like 3-SAT or Bicoloring it provides new upper bounds for their critical thresholds (γ_c^{var}=4.396 and γ_c^{var}=2.149 resp.).

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S. Franz, M. Leone, F. Ricci-Tersenghi, R. Zecchina. 2001-08-15. Exact solutions for diluted spin glasses and optimization problems. https://doi.org/10.1103/physrevlett.87.127209

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