Probabilistic Methods for Algorithmic Discrete Mathematics (Algorithms and Combinatorics (16))

The book gives an accessible account of modern pro- babilistic methods for analyzing combinatorial structures and algorithms. Each topic is approached in a didactic manner but the most recent developments are linked to the basic ma- terial. Extensive lists of references and a detailed index will make this a useful guide for graduate students and researchers. Special features included: - a simple treatment of Talagrand inequalities and their applications - an overview and many carefully worked out examples of the probabilistic analysis of combinatorial algorithms - a discussion of the "exact simulation" algorithm (in the context of Markov Chain Monte Carlo Methods) - a general method for finding asymptotically optimal or near optimal graph colouring, showing how the probabilistic method may be fine-tuned to explit the structure of the underlying graph - a succinct treatment of randomized algorithms and derandomization techniques. Leave nothing to chance. This cliche embodies the common belief that ran­ domness has no place in carefully planned methodologies, every step should be spelled out, each i dotted and each t crossed. In discrete mathematics at least, nothing could be further from the truth. Introducing random choices into algorithms can improve their performance. The application of proba­ bilistic tools has led to the resolution of combinatorial problems which had resisted attack for decades. The chapters in this volume explore and celebrate this fact. Our intention was to bring together, for the first time, accessible discus­ sions of the disparate ways in which probabilistic ideas are enriching discrete mathematics. These discussions are aimed at mathematicians with a good combinatorial background but require only a passing acquaintance with the basic definitions in probability (e.g. expected value, conditional probability). A reader who already has a firm grasp on the area will be interested in the original research, novel syntheses, and discussions of ongoing developments scattered throughout the book. Some of the most convincing demonstrations of the power of these tech­ niques are randomized algorithms for estimating quantities which are hard to compute exactly. One example is the randomized algorithm of Dyer, Frieze and Kannan for estimating the volume of a polyhedron. To illustrate these techniques, we consider a simple related problem. Suppose S is some region of the unit square defined by a system of polynomial inequalities: Pi (x. y) ~ o.

The book gives an accessible account of modern probabilistic methods for analyzing combinatorial structures and algorithms. It will be an useful guide for graduate students and researchers.
Special features included: a simple treatment of Talagrand's inequalities and their applications; an overview and many carefully worked out examples of the probabilistic analysis of combinatorial algorithms; a discussion of the "exact simulation" algorithm (in the context of Markov Chain Monte Carlo Methods); a general method for finding asymptotically optimal or near optimal graph colouring, showing how the probabilistic method may be fine-tuned to exploit the structure of the underlying graph; a succinct treatment of randomized algorithms and derandomization techniques.

Front Matter....Pages I-XVII The Probabilistic Method....Pages 1-35 Probabilistic Analysis of Algorithms....Pages 36-92 An Overview of Randomized Algorithms....Pages 93-115 Mathematical Foundations of the Markov Chain Monte Carlo Method....Pages 116-165 Percolation and the Random Cluster Model: Combinatorial and Algorithmic Problems....Pages 166-194 Concentration....Pages 195-248 Branching Processes and Their Applications in the Analysis of Tree Structures and Tree Algorithms....Pages 249-314 Back Matter....Pages 315-325 An Overview of Randomized Algorithms by R.Motwani, P.Raghavan Concentration by C.McDiarmid The Probabilistic Method by M.Molloy Percolation and the Random Cluster Model: Combinatorial and Algorithmic Problems by D.Welsh Mathematical Foundations of the Markov Chain Monte Carlo Method by M.Jerrum Probabilistic Analysis of Algorithms by A.M.Frieze, B.Reed Branching Processes and Their Applications in the Analysis of Tree Structures and Tree Algorithms by L.Devroye.