Randomized Algorithms: Approximation, Generation and Counting

Randomized Algorithms: Approximation, Generation and Counting (Distinguished Dissertations)



This study discusses two problems of fine pedigree - counting and generation - both of which are of fundamental importance to discrete mathematics and probability. When asking questions like "How many are there?" and "What does it look like on average?" of families of combinatorial structures, answers are often difficult to find - we can be blocked by seemingly intractable algorithms. This text shows how to get around the problem of intractability with the Markov chain Monte Carlo method, as well as highlighting the method's natural limits. It uses the technique of coupling before introducing "path coupling", a technique which radically simplifies and improves upon other methods in the area.