Title Hamiltonian cycle problem and markov chains / Vivek S. Borkar [and others]

Published New York, NY : Springer, ©2012

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Description 1 online resource (xiv, 201 pages) Series International series in operations research & management science, 0884-8289 ; v. 171 International series in operations research & management science ; v. 171. Contents Introduction -- Hopf Bifurcation and Normal Form Computation -- Comparison of Methods for Computing Focus Values -- Application (I)-Hilbert's 16th Problem -- Application (II)-Practical Problems -- Fundamental Theory of the Melnikov Function Method -- Limit Cycle Bifurcations Near a Center -- Limit Cycles Near a Homoclinic or Heteroclinic Loop -- Finding More Limit Cycles Using Melnikov Functions -- Limit Cycle Bifurcations in Equivariant Systems Summary 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 the reported results is achieved by assigning probabilistic interpretation to key elements of the original deterministic problems. In particular, approaches summarized here build on a technique that embeds Hamiltonian Cycle and Traveling 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 more than fifteen research papers appearing in journals catering to disparate audiences such as:¡ MOR, Random Structures and Algorithms, SIAM J. on Discrete Mathematics, Optimization, ¡ J. of Mathematical Analysis and Applications and some others. Furthermore, because of the evolution of this topic and specific orientation of these journals, the published manuscripts are often written in a very terse manner and use disparate notation.¡ As such it is difficult for new researchers to make use of the many advances reported in these papers. Hence the main purpose of this book is to present a concise and yet, well written, synthesis of the majority of the theoretical and algorithmic results obtained so far.¡ In addition the book will discuss numerous open questions and problems that arise from this body of work and which are yet to be fully solved.¡ The authors believe that their approach casts the Hamiltonian Cycle and Traveling Salesman problems in a mathematical framework that permits analytical concepts and techniques, ¡ not used hitherto in their context, to be brought to bear to further clarify both the underlying difficulty of NP-completeness of these problems 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 but, unlike the research manuscripts where they were originally proved, ¡ are illustrated with numerous examples Analysis Economics Mathematical optimization Distribution (Probability theory) Economics/Management Science Operations Research/Decision Theory Operations Research, Management Science Optimization Probability Theory and Stochastic Processes economie bedrijfswetenschap management science waarschijnlijkheidstheorie probability theory stochastische processen stochastic processes operationeel onderzoek operations research optimalisatie Management studies, Business Administration, Organizational Science (General) Economics (General) Management, bedrijfskunde, organisatiekunde (algemeen) Economie (algemeen) Bibliography Includes bibliographical references and index Subject Hamiltonian systems. Markov processes. MATHEMATICS -- Differential Equations -- General. Science économique. Affaires. Hamiltonian systems Markov processes Form Electronic book Author Borkar, Vivek S ISBN 9781461432326 1461432324

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