Leave nothing to chance. This cliche embodies the common belief that randomness has no place in carefully planned methodologies, every step should be spelled put, 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 probabilistic 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 discussions 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 techniques 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: р<(х,у) < 0. Then the area of S is equal to the probability that a random point is in S, where the point is chosen uniformly at random from the unit square. Furthermore, we can determine if a point is in S simply by evaluating each polynomial at this point. So, we can estimate the area of 5 by the proportion of a sufficently large set of-random points which lie in S. For this problem, choosing a random sample point was straightforward, as was using the sample to estimate the area. Estimating the volume of a polyhedron is not so simple.
The central chapter in this volume was written by Jerrum. It discusses more sophisticated techniques for generating random sample points from a probability distribution and using them to develop randomized algorithms for approximate counting. In particular, he discusses techniques for showing
