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The subject of these notes is counting (of combinatorial structures) and related topics, viewed from a computational perspective. "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).
A major theme of the book is the idea of accumulating information about a set of combinatorial structures by performing a random walk (i.e., simulating a Markov chain) on those structures. (This is for the discrete setting; one can also learn about a geometric body by performing a walk within it.) The running time of such an algorithm depends on the rate of convergence to equilibrium of this Markov chain, as formalised in the notion of "mixing time" of the Markov chain. A significant proportion of the volume is given over to an investigation of techniques for bounding the mixing time in cases of computational interest.
These notes will be of value not only to teachers of postgraduate courses on these topics, but also to established researchers in the field of computational complexity who wish to become acquainted with recent work on non-asymptotic analysis of Markov chains, and their counterparts in stochastic processes who wish to discover how their subject sits within a computational context. For the first time this body of knowledge has been brought together in a single volume.
Content Level »Research
Keywords »Markov chain - Matchings - Probability theory - algorithms - complexity - computer science - mixing - probability - random walk - sets - statistical physics
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.