Counting, Sampling and Integrating Algorithms and Complexity [Paperback]

The subject of these notes is counting and related topics, viewed from a computational perspective. A major theme of the book is the idea of accumulating information about a set of combinatorial structures by performing a random walk on those structures. These notes will be of value not only to teachers of postgraduate courses on these topics, but also to established researchers. For the first time this body of knowledge has been brought together in a single volume.

These notes had their origin in a postgraduate lecture series I gave at the Eid? genossiche Technische Hochschule (ETH) in Zurich in the Spring of 2000. I am very grateful to my hosts, the Forschungsinstitut fUr Mathematik at ETH, for providing the ideal opportunity to develop and present this material in what I hope is a reasonably coherent manner, and also for encouraging and assisting me to record the proceedings in these lecture notes. The subject of the lecture series was counting (of combinatorial structures) and related topics, viewed from a computational perspective. As we shall see, related topics include sampling combinatorial structures (being computationally equivalent to approximate counting via efficient reductions), evaluating partition functions (being weighted counting) and calculating the volume of bodies (being counting in the limit). We shall be inhabiting a different world to the one conjured up by books with titles like Combinatorial Enumeration or Graphical Enumeration. There, the prob? lems are usually parameterised on a single integer parameter n, and the required solutions are closed form or asymptotic estimates obtained using very refined and precise analytical tools.Foreword.- 1 Two good counting algorithms.- 1.1 Spanning trees.- 1.2 Perfect matchings in a planar graph.- 2 #P-completeness.- 2.1 The class #P.- 2.2 A primal #P-complete problem.- 2.3 Computing the permanent is hard on average.- 3 Sampling and counting.- 3.1 Preliminaries.- 3.2 Reducing approxilS2