Counting, Sampling and Integrating: Algorithms and Complexity / Edition 1

by Mark Jerrum

Overview

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 problems 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.

Product Details

ISBN-13:	9783764369460
Publisher:	Birkhäuser Basel
Publication date:	04/28/2003
Series:	Lectures in Mathematics. ETH Zürich
Edition description:	2003
Pages:	112
Product dimensions:	7.01(w) x 10.00(h) x 0.01(d)

Table of Contents

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 approximate countingto almost uniform sampling.- 3.3 Markov chains.- 4 Coupling and colourings.- 4.1 Colourings of a low-degree graph.- 4.2 Bounding mixing time using coupling.- 4.3 Path coupling.- 5 Canonical paths and matchings.- 5.1 Matchings in a graph.- 5.2 Canonical paths.- 5.3 Back to matchings.- 5.4 Extensions and further applications.- 5.5 Continuous time.- 6 Volume of a convex body.- 6.1 A few remarks on Markov chainswith continuous state space.- 6.2 Invariant measure of the ball walk.- 6.3 Mixing rate of the ball walk.- 6.4 Proof of the Poincarü inequality (Theorem 6.7).- 6.5 Proofs of the geometric lemmas.- 6.6 Relaxing the curvature condition.- 6.7 Using samples to estimate volume.- 6.8 Appendix: a proof of Corollary 6.8.- 7 Inapproximability.- 7.1 Independent sets in a low degree graph.