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Counting, Sampling and Integrating: Algorithm and Complexity
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Feldname
Details
Dewey Class
519.2
Titel
Counting, Sampling and Integrating: Algorithm and Complexity ([EBook] /) / by Mark Jerrum.
Verfasser
Jerrum, Mark
Other name(s)
SpringerLink (Online service)
Veröffentl
Basel : : Birkhäuser Basel, , 2003.
Physical Details
XI, 112 p. 7 illus. : online resource.
Reihe
Lectures in mathematics / ETH Zürich
ISBN
9783034880053
Summary Note
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.:
Contents note
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.
System details note
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site
http://dx.doi.org/10.1007/978-3-0348-8005-3
LINKS ZU 'VERWANDTEN WERKEN
Schlagwörter:
.
Dynamical Systems
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Mathematical Statistics
.
Mathematics
.
Probabilities
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Probability and Statistics in Computer Science
.
Probability theory and stochastic processes
.
Statistical Physics
.
Statistical Physics, Dynamical Systems and Complexity
.
Authors:
Jerrum, Mark
.
Corporate Authors:
SpringerLink (Online service)
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Series:
Lectures in mathematics / ETH Zürich
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Classification:
519.2
.
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