International Series in Operations Research & Management Science

Hamiltonian Cycle Problem and Markov Chains

Authors: Borkar, V.S., Ejov, V., Filar, J.A., Nguyen, G.T.

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  • Concise, readable synthesis of important yet scattered research to date
  • Discusses problems yet to be fully solved
  • Provides many examples not supplied in original studies
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About this book

This research monograph summarizes a line of research that maps certain classical problems of discrete mathematics and operations research - such as the Hamiltonian Cycle and the Travelling Salesman Problems - into convex domains where continuum analysis can be carried out. Arguably, the inherent difficulty of these, now classical, problems stems precisely from the discrete nature of domains in which these problems are posed. The convexification of domains underpinning these results is achieved by assigning probabilistic interpretation to key elements of the original deterministic problems. In particular, the approaches summarized here build on a technique that embeds Hamiltonian Cycle and Travelling Salesman Problems in a structured singularly perturbed Markov decision process. The unifying idea is to interpret subgraphs traced out by deterministic policies (including Hamiltonian cycles, if any) as extreme points of a convex polyhedron in a space filled with randomized policies.

The above innovative approach has now evolved to the point where there are many, both theoretical and algorithmic, results that exploit the nexus between graph theoretic structures and both probabilistic and algebraic entities of related Markov chains. The latter include moments of first return times, limiting frequencies of visits to nodes, or the spectra of certain matrices traditionally associated with the analysis of Markov chains. However, these results and algorithms are dispersed over many research papers appearing in journals catering to disparate audiences. As a result, the published manuscripts are often written in a very terse manner and use disparate notation, thereby making it difficult  for new researchers to make use of the many reported advances.

Hence the main purpose of this book is to present a concise and yet easily accessible synthesis of the majority of the theoretical and algorithmic results obtained so far. In addition, the book discusses numerous open questions and problems that arise from this body of work and which are yet to be fully solved. The approach casts the Hamiltonian Cycle Problem in a mathematical framework that permits analytical concepts and techniques, not used hitherto in this context, to be brought to bear to further clarify both the underlying difficulty of NP-completeness of this problem and the relative exceptionality of truly difficult instances. Finally, the material is arranged in such a manner that the introductory chapters require very little mathematical background and discuss instances of graphs with interesting structures that motivated a lot of the research in this topic. More difficult results are introduced later  and are illustrated with numerous examples.

Table of contents (10 chapters)

Table of contents (10 chapters)

Buy this book

eBook 107,09 €
price for Spain (gross)
  • ISBN 978-1-4614-3232-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover 135,19 €
price for Spain (gross)
  • ISBN 978-1-4614-3231-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
Softcover 135,19 €
price for Spain (gross)
  • ISBN 978-1-4899-9227-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
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Bibliographic Information

Bibliographic Information
Book Title
Hamiltonian Cycle Problem and Markov Chains
Authors
Series Title
International Series in Operations Research & Management Science
Series Volume
171
Copyright
2012
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media, LLC
eBook ISBN
978-1-4614-3232-6
DOI
10.1007/978-1-4614-3232-6
Hardcover ISBN
978-1-4614-3231-9
Softcover ISBN
978-1-4899-9227-7
Series ISSN
0884-8289
Edition Number
1
Number of Pages
XIV, 202
Topics