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Hamiltonian Cycle Problem and Markov Chains (Hardcover, 2012 ed.)
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Hamiltonian Cycle Problem and Markov Chains (Hardcover, 2012 ed.)
Series: International Series in Operations Research & Management Science, 171
Expected to ship within 10 - 15 working days
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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.
General
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