We study the graph coloring problem over random graphs of finite average connectivity c. Given a number q of available colors, we find that graphs with low connectivity admit almost always a proper coloring, whereas graphs with high connectivity are uncolorable. Depending on q, we find the precise value of the critical average connectivity c(q). Moreover, we show that below c(q) there exists a clustering phase c in [c(d),c(q)] in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms.

We study the graph coloring problem over random graphs of finite average connectivity c. Given a number q of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on q, we find with a one-step replica-symmetry breaking approximation the precise value of the critical average connectivity c(q). Moreover, we show that below c(q) there exists a clustering phase c in [c(d),c(q)] in which ground states spontaneously divide into an exponential number of clusters. Furthermore, we extended our considerations to the case of single instances showing consistent results. This leads us to propose a different algorithm that is able to color in polynomial time random graphs in the hard but colorable region, i.e., when c in [c(d),c(q)].

We present a general formalism to make the Replica-Symmetric and Replica-Symmetry-Breaking ansatz in the context of Kikuchi's Cluster Variational Method (CVM). Using replicas and the message-passing formulation of CVM we obtain a variational expression of the replicated free energy of a system with quenched disorder, both averaged and on a single sample, and make the hierarchical ansatz using functionals of functions of fields to represent the messages. We obtain a set of integral equations for the message functionals. The main difference with the Bethe case is that the functionals appear in the equations in implicit form and are not positive definite, thus standard iterative population dynamic algorithms cannot be used to determine them. In the simplest cases the solution could be obtained iteratively using Fourier transforms. We begin to study the method considering the plaquette approximation to the averaged free energy of the Edwards-Anderson model in the paramagnetic Replica-Symmetric phase. In two dimensions we find that the spurious spin-glass phase transition of the Bethe approximation disappears and the paramagnetic phase is stable down to zero temperature on the square lattice for different random interactions. The quantitative estimates of the free energy and of various other quantities improve those of the Bethe approximation. The plaquette approximation fails to predict a second-order spin-glass phase transition on the cubic 3D lattice but yields good results in dimension four and higher. We provide the physical interpretation of the beliefs in the replica-symmetric phase as disorder distributions of the local Hamiltonian. The messages instead do not admit such an interpretation and indeed they cannot be represented as populations in the spin-glass phase at variance with the Bethe approximation. The approach can be used in principle to study the phase diagram of a wide range of disordered systems and it is also possible that it can be used to get quantitative predictions on single samples. These further developments present however great technical challenges

We establish a general mechanism for highly efficient quantum transport through finite, disordered 3D networks. It relies on the interplay of disorder with centrosymmetry and a dominant doublet spectral structure and can be controlled by the proper tuning of only coarse-grained quantities. Photosynthetic light harvesting complexes are discussed as potential biological incarnations of this design principle.