The Metropolis Algorithm has been the most successful and influential of all the members of the computational species that used to be called the "Monte Carlo Method. " Today, topics related to this algorithm constitute an entire field of computational science supported by a deep theory and having applications ranging from physical simulations to the foundations of computational complexity. he story goes that Stan Ulam was in a Los Angclcs hospital recupcrating and, to stavc off horedoni, he tried computing the probability of getting a "perfect" solitaire hand. Bcfnre long, he hit on the idea of using random sampling: Choose a solitaire hand a t random. If it is pcrfect, let count = cnzint + 1; if not, let count = count. Aftcr M san
Our starting point is an algorithm of Kenyon, Randall, and Sinclair, which is built upon the ideas of Jerrum and Sinclair, giving an approximation to crucial parameters of the monomer-dimer covering problem in polynomial time. We make two key improvements to their algorithm: we greatly reduce the number of simulations that must be run by estimating good values of the generating function parameter, and we greatly reduce the number of steps that must be taken in each simulation by aggregating to a simulation with at most five states. The result is an algorithm that is computationally feasible for modestly sized meshes. We use our algorithm on two- and three-dimensional problems, computing approximations to the coefficients of the generating function and some limiting values.
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