
Counting, Sampling and Integrating: Algorithms and Complexity
Mark Jerrum(Author)
Birkhäuser (Publisher)
Published on 23. January 2003
Book
Paperback/Softback
XI, 112 pages
978-3-7643-6946-0 (ISBN)
Description
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.
More details
Series
Edition
2003 ed.
Language
English
Place of publication
Basel
Switzerland
Publishing group
Springer Basel
Target group
Professional and scholarly
Research
Illustrations
7 s/w Abbildungen
XI, 112 p. 7 illus.
Dimensions
Height: 254 mm
Width: 178 mm
Thickness: 8 mm
Weight
257 gr
ISBN-13
978-3-7643-6946-0 (9783764369460)
DOI
10.1007/978-3-0348-8005-3
Schweitzer Classification
Content
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.