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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.
Over the past decade, many major advances have been made in the field of graph colouring via the probabilistic method. This monograph provides an accessible and unified treatment of these results, using tools such as the Lovasz Local Lemma and Talagrand's concentration inequality.The topics covered include: Kahn's proofs that the Goldberg-Seymour and List Colouring Conjectures hold asymptotically; a proof that for some absolute constant C, every graph of maximum degree Delta has a Delta+C total colouring; Johansson's proof that a triangle free graph has a O(Delta over log Delta) colouring; algorithmic variants of the Local Lemma which permit the efficient construction of many optimal and near-optimal colourings.This begins with a gentle introduction to the probabilistic method and will be useful to researchers and graduate students in graph theory, discrete mathematics, theoretical computer science and probability.
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.
The author writes of his colorful life providing strippers for
mob-owned strip joints in Philadelphia, leaving town to become a
Las Vegas casino dealer, and then returning to Philadelphia to work
for more than 20 years in law enforcement.
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