Rémi Monasson

Determining computational complexity from characteristic 'phase transitions'

Non-deterministic polynomial time (commonly termed ‘NP-complete’) problems are relevant to many computational tasks of practical interest—such as the ‘travelling salesman problem’—but are difficult to solve: the computing time grows exponentially with problem size in the worst case. It has recently been shown that these problems exhibit ‘phase boundaries’, across which dramatic changes occur in the computational difficulty and solution character—the problems become easier to solve away from the boundary. Here we report an analytic solution and experimental investigation of the phase transition in K -satisfiability, an archetypal NP-complete problem. Depending on the input parameters, the computing time may grow exponentially or polynomially with problem size; in the former case, we observe a discontinuous transition, whereas in the latter case a continuous (second-order) transition is found. The nature of these transitions may explain the differing computational costs, and suggests directions for improving the efficiency of search algorithms. Similar types of transition should occur in other combinatorial problems and in glassy or granular materials, thereby strengthening the link between computational models and properties of physical systems.

Diffusion, localization and dispersion relations on “small-world” lattices

Abstract:The spectral properties of the Laplacian operator on “small-world” lattices, that is mixtures of unidimensional chains and random graphs structures are investigated numerically and analytically. A transfer matrix formalism including a self-consistent potential à la Edwards is introduced. In the extended region of the spectrum, an effective medium calculation provides the density of states and pseudo relations of dispersion for the eigenmodes in close agreement with the simulations. Localization effects, which are due to connectivity fluctuations of the sites are shown to be quantitatively described by the single defect approximation recently introduced for random graphs.

Structural glass transition and the entropy of the metastable states.

The metastable states of a glass are counted by adding a weak pinning field which explicitly breaks the ergodicity. Their entropy, that is the logarithm of their number, is extensive in a range of temperatures

Statistical mechanics methods and phase transitions in optimizationproblems

T_G < T < T_C

Neuronal couplings between retinal ganglion cells inferred by efficient inverse statistical physics methods

only, where

Force and kinetic barriers to unzipping of the DNA double helix

T_G

Small-correlation expansions for the inverse Ising problem

and

Statistical Mechanics of Torque Induced Denaturation of DNA

T_C

Entropy of the K -Satisfiability Problem

correspond to the ideal calorimetric and kinetic glass transition temperatures respectively. An explicit self-consistent computation of the metastable states entropy for a non disordered model is given.

Statistical mechanics of the random K -satisfiability model

Recently, it has been recognized that phase transitions play an important role in the probabilistic analysis of combinatorial optimization problems. However, there are in fact many other relations that lead to close ties between computer science and statistical physics. This review aims at presenting the tools and concepts designed by physicists to deal with optimization or decision problems in a language accessible for computer scientists and mathematicians, with no prerequisites in physics. We first introduce some elementary methods of statistical mechanics and then progressively cover the tools appropriate for disordered systems. In each case, we apply these methods to study the phase transitions or the statistical properties of the optimal solutions in various combinatorial problems. We cover in detail the Random Graph, the Satisfiability, andthe Traveling Salesman problems. References to the physics literature on optimization are provided. We also give our perspective regarding the interdisciplinary contribution of physics to computer science.