Randomized Algorithms: Approximation, Generation, and Counting [Paperback]

Randomized Algorithms 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. Randomized Algorithms 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 new technique which radically simplifies and improves upon previous methods in the area.Randomized Algorithms 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. Randomized Algorithms 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 new technique which radically simplifies and improves upon previous methods in the area.1 Mathematical Background.- 1.1 Computational Complexity.- 1.1.1 Introduction.- 1.1.2 Notation for Asymptotics: ?, ?, and ?.- 1.1.3 Complexity Classes.- 1.1.4 Oracles, Reductions, and Hardness.- 1.1.5 More Complexity Classes.- 1.1.6 Conclusion.- 1.1.7 Some Common Problems in Complexity Theory.- 1.2 Probability.- 1.3 Markov Chains.- 1.4 Graph Theory.- 1.4.1 TutteGr?thendieck Polynomial.- 1.4.2 Hypergraphs.- 2 Techniques for Sampling and Approximate Sampling.- 2.1 l³’