Exact solutions for diluted spin glasses and optimization problems

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, gamma (d) = 0.818, and gamma (s) = 0.918), whereas for hard optimization problems like 3-SAT or Bicoloring it provides new upper bounds for their critical thresholds (gamma (var)(c) = 4.396 and gamma (var)(c) = 2.149).