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 $p=3\mathrm{}$, ${\gamma }_d=0.818\mathrm{}$, and ${\gamma }_s=0.918\mathrm{}$), whereas for hard optimization problems like 3-SAT or Bicoloring it provides new upper bounds for their critical thresholds ( ${\gamma }_c^{\mathrm{var}}=4.396\mathrm{}$ and ${\gamma }_c^{\mathrm{var}}=2.149\mathrm{}$).

• Received 16 March 2001

DOI:https://doi.org/10.1103/PhysRevLett.87.127209