Counting, Sampling and Integrating: Algorithms and Complexity

1. Front Matter

   Pages i-xi

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2. Two good counting algorithms

   • Mark Jerrum

   Pages 1-10

3. #P-completeness

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   Pages 11-23

4. Sampling and counting

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   Pages 25-32

5. Coupling and colourings

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   Pages 33-47

6. Canonical paths and matchings

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   Pages 49-64

7. Volume of a convex body

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   Pages 65-92

8. Inapproximability

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   Pages 93-103

9. Back Matter

   Pages 105-112

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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.

• Markov chain
• Matchings
• Probability theory
• algorithms
• complexity
• computer science
• mixing
• probability
• random walk
• sets
• statistical physics

• Division of Informatics, The University of Edinburgh, Edinburgh, Scotland, UK

  Mark Jerrum

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