Controlled markov chains, graphs and hamiltonicity
By: Jerzy A. Filar
Publisher: Now Publishers
Published: 2007
Place of publication: Hanover, Mass
Language: eng
Format: [electronic resource] /
Number of pages: 85
ISBN-13: 9781601980892
Subjects: Probability & Statistics; MATHEMATICS; Hamiltonian graph theory; Stochastic Processes; Markov processes
Other titles: Foundations and trends in stochastic systems.
Description: This manuscript summarizes a line of research that maps certain classical problems of discrete mathematics -- such as the Hamiltonian Cycle and the Traveling 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 topic 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. Numerous open questions and problems are described in the presentation.
Notes: Offprint of Foundations and trends in stochastic systems. vol. 1, no. 2, p. 77-162, 2006.

Title from PDF t.p. (viewed July 7, 2008).

Includes bibliographical references (p. 81-85).
Table of contents: title: Abstract; title: 1. Embedding of a graph in a Markov decision process; title: 2. Analysis in the policy space; title: 3. Analysis in the frequency space; title: 4. Spectral properties, spin-offs, and speculation; title: Acknowledgments; title: References.