Algorithms in Monte Carlo Methods

Dear Colleagues,

Monte Carlo methods, in the most general sense, encompass algorithms that can be characterized by two key features: (1) they use random choices in their operation; and (2) they relax the requirement of insisting on a deterministically correct result. A Monte Carlo method may produce a solution that is correct only with high probability (but not with certainty), with respect to the internal random choices of the algorithm.

Such an algorithm can have a variety of goals. It may solve a deterministic problem, for example, finding a minimum cut in a graph. It may look for an approximate solution of a numerical task, including optimization and numerical integration. The target can also be inherently random, such as drawing a random sample from a complicated probability distribution.

The key issue in designing a Monte Carlo method is to find an algorithm that can capitalize on the random choices to efficiently produce a result, which is correct with high probability. In some cases, such an algorithm is completely problem specific; examples include the Karger–Stein minimum cut algorithm and various randomized primality tests. However, there are useful general schemes which provide solutions to a multitude of tasks. Quite a few such schemes exist. These include general-purpose randomized optimization methods such as simulated annealing, genetic algorithm, solutions for optimal stopping, change-point detection, the Moser–Tardos algorithm, etc. Another class of methods aim at producing random samples from complex probability distributions, including rejection sampling, importance sampling, the cross-entropy method, Markov chain Monte Carlo algorithms, and others.

This is a broad and rich research area. Topics of interest include, but are not limited to, the following:

• Randomized algorithms for graph problems and combinatorial optimization;
• Randomized algorithms for continuous optimization;
• Randomized algorithms for numerical problems;
• Methods for generating random samples from complex probability distributions;
• Optimal stopping rules;
• Change-point detection methods;
• Markov chain Monte Carlo algorithms;
• Estimation of mixing rate in Markov chains;
• Computational complexity issues relating to Monte Carlo methods;
• Case studies of solving practical problems via Monte Carlo methods.

Dr. Georgy Sofronov
Prof. Dr. Andras Farago
Guest Editors

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