Algorithms for Random Generation and Counting: A Markov Chain Approach

A. Sinclair (Autor)

Buch | Hardcover

147 Seiten

1993
Birkhauser Boston Inc (Verlag)
978-0-8176-3658-6 (ISBN)

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This monograph is a slightly revised version of my PhD thesis [86], com pleted in the Department of Computer Science at the University of Edin burgh in June 1988, with an additional chapter summarising more recent developments. Some of the material has appeared in the form of papers [50,88]. The underlying theme of the monograph is the study of two classical problems: counting the elements of a finite set of combinatorial structures, and generating them uniformly at random. In their exact form, these prob lems appear to be intractable for many important structures, so interest has focused on finding efficient randomised algorithms that solve them ap proxim~ly, with a small probability of error. For most natural structures the two problems are intimately connected at this level of approximation, so it is natural to study them together. At the heart of the monograph is a single algorithmic paradigm: sim ulate a Markov chain whose states are combinatorial structures and which converges to a known probability distribution over them. This technique has applications not only in combinatorial counting and generation, but also in several other areas such as statistical physics and combinatorial optimi sation. The efficiency of the technique in any application depends crucially on the rate of convergence of the Markov chain.

Synopsis.- 1 Preliminaries.- 1.1 Some basic definitions.- 1.2 Notions of tractability.- 1.3 An extended model.- 1.4 Counting, generation and self-reducibility.- 1.5 An interesting class of relations.- 2 Markov chains and rapid mixing.- 2.1 The Markov chain approach to generation problems.- 2.2 Conductance and the rate of convergence.- 2.3 A characterisation of rapid mixing.- 3 Direct Applications.- 3.1 Some simple examples.- 3.2 Approximating the permanent.- 3.3 Monomer-dimer systems.- 3.4 Concluding remarks.- 4 Indirect Applications.- 4.1 A robust notion of approximate counting.- 4.2 Self-embeddable relations.- 4.3 Graphs with specified degrees.- Appendix: Recent developments.

Reihe/Serie	Progress in Theoretical Computer Science
Zusatzinfo	VIII, 147 p.
Verlagsort	Secaucus
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Informatik ► Theorie / Studium
Themenwelt	Mathematik / Informatik ► Mathematik
ISBN-10	0-8176-3658-7 / 0817636587
ISBN-13	978-0-8176-3658-6 / 9780817636586
Zustand	Neuware