Vertex cover problem studied by cavity method: Analytics and population dynamics

We study the vertex cover problem on finite connectivity random graphs by zero-temperature cavity method. The minimum vertex cover corresponds to the ground state(s) of a proposed Ising spin model. When the connectivity c>e=2.718282, there is no state for this system as the reweighting parameter y, which takes a similar role as the inverse temperature \beta in conventional statistical physics, approaches infinity; consequently the ground state energy is obtained at a finite value of y when the free energy function attains its maximum value. The minimum vertex cover size at given c is estimated using population dynamics and compared with known rigorous bounds and numerical results. The backbone size is also calculated.Comment: 7 pages (including 3 figures and 1 table), REVTeX4 forma

On Efficient Second Order Stabilized Semi-Implicit Schemes for the Cahn-Hilliard Phase-Field Equation

Efficient and energy stable high order time marching schemes are very important but not easy to construct for the study of nonlinear phase dynamics. In this paper, we propose and study two linearly stabilized second order semi-implicit schemes for the Cahn-Hilliard phase-field equation. One uses backward differentiation formula and the other uses Crank-Nicolson method to discretize linear terms. In both schemes, the nonlinear bulk forces are treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic systems with constant coefficients, for which lots of robust and efficient solvers are available. The discrete energy dissipation properties are proved for both schemes. Rigorous error analysis is carried out to show that, when the time step-size is small enough, second order accuracy in time is obtained with a prefactor controlled by a fixed power of $1/\varepsilon$, where $\varepsilon$ is the characteristic interface thickness. Numerical results are presented to verify the accuracy and efficiency of proposed schemes

Combined local search strategy for learning in networks of binary synapses

Learning in networks of binary synapses is known to be an NP-complete problem. A combined stochastic local search strategy in the synaptic weight space is constructed to further improve the learning performance of a single random walker. We apply two correlated random walkers guided by their Hamming distance and associated energy costs (the number of unlearned patterns) to learn a same large set of patterns. Each walker first learns a small part of the whole pattern set (partially different for both walkers but with the same amount of patterns) and then both walkers explore their respective weight spaces cooperatively to find a solution to classify the whole pattern set correctly. The desired solutions locate at the common parts of weight spaces explored by these two walkers. The efficiency of this combined strategy is supported by our extensive numerical simulations and the typical Hamming distance as well as energy cost is estimated by an annealed computation.Comment: 7 pages, 4 figures, figures and references adde