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Proof of the Hamiltonicity-Trace Conjecture for Singularly Perturbed Markov Chains

Part of: Markov processes Polynomials and matrices Graph theory

Published online by Cambridge University Press:  14 July 2016

Vladimir Ejov ,
Nelly Litvak ,
Giang T. Nguyen  and
Peter G. Taylor
Vladimir Ejov*
Affiliation:
University of South Australia
Nelly Litvak*
Affiliation:
University of Twente
Giang T. Nguyen*
Affiliation:
University of South Australia
Peter G. Taylor*
Affiliation:
University of Melbourne
*
∗ Postal address: School of Mathematics and Statistics, University of South Australia, Mawson Lakes campus, Mawson Lakes SA 5095, Australia. Email address: ejovvl@gmail.com
∗∗ Postal address: Faculty of Electrical Engineering, Mathematics and Computer Science, Department of Applied Mathematics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands. Email address: n.litvak@ewi.utwente.nl
∗∗∗ Current address: Département d'informatique, Université Libre de Bruxelles, CP 212, Boulevard du Triomphe, 2, B-1050 Bruxelles, Belgium. Email address: giang.nguyen@ulb.ac.be
∗∗∗∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia. Email address: p.taylor@ms.unimelb.edu.au
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Abstract

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We prove the conjecture formulated in Litvak and Ejov (2009), namely, that the trace of the fundamental matrix of a singularly perturbed Markov chain that corresponds to a stochastic policy feasible for a given graph is minimised at policies corresponding to Hamiltonian cycles.

Keywords

Stochastic matrixHamiltonian cycleperturbed Markov chain

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 05C45: Eulerian and Hamiltonian graphs 11C20: Matrices, determinants
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 901 - 910
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Andramonov, M., Filar, J., Pardalos, P. and Rubinov, A. (2000). Hamiltonian cycle problem via Markov chains and min-type approaches. In Approximation and Complexity in Numerical Optimization, ed. Pardalos, P., Kluwer, Dordrecht, pp. 31–47.Google Scholar
[2] Borkar, V. S., Ejov, V. and Filar, J. A. (2004). Directed graphs, Hamiltonicity and doubly stochastic matrices. Random Structures Algorithms 25, 376–395.Google Scholar
[3] Borkar, V. S., Ejov, V. and Filar, J. A. (2009). On the Hamiltonicity gap and doubly stochastic matrices. Random Structures Algorithms 34, 502–519.Google Scholar
[4] Brin, S. and Page, L. (1998). The anatomy of a large-scale hypertextual Web search engine. Comput. Networks ISDN Systems 30, 107–117.Google Scholar
[5] Ejov, V. and Nguyen, G. T. (2009). Consistent behavior of certain perturbed determinants induced by graphs. Linear Algebra Appl. 431, 543–552.Google Scholar
[6] Ejov, V., Filar, J. and Gondzio, J. (2004). An interior point heuristic for the Hamiltonian cycle problem via Markov decision processes. J. Global Optimization 29, 315–334.Google Scholar
[7] Ejov, V., Filar, J. A. and Nguyen, M.-T. (2004). Hamiltonian cycles and singularly perturbed Markov chains. Math. Operat. Res. 29, 114–131.Google Scholar
[8] Ejov, V., Filar, J. A., Haythorpe, M. and Nguyen, G. T. (2009). Refined MDP-based branch-and-fix algorithm for the Hamiltonian cycle problem. Math. Operat. Res. 34, 758–768.Google Scholar
[9] Ejov, V., Filar, J., Murray, W. and Nguyen, G. T. (2008). Determinants and longest cycles of graphs. SIAM J. Discrete Math. 22, 1215–1225.Google Scholar
[10] Ejov, V., Litvak, N., Nguyen, G. T. and Taylor, P. G. (2008). Proof of the Hamiltonicity-trace conjecture for singularly perturbed Markov chains. Preprint.Google Scholar
[11] Filar, J. A. and Krass, D. (1994). Hamiltonian cycles and Markov chains. Math. Operat. Res. 19, 223–237.Google Scholar
[12] Filar, J. A. and Lasserre, J. B. (2000). A non-standard branch and bound method for the Hamiltonian cycle problem. ANZIAM J. 42, C586–C607.Google Scholar
[13] Grinstead, C. M. and Snell, J. L. (1997). Introduction to Probability, 2nd edn. American Mathematical Society, Providence, RI.Google Scholar
[14] Haythorpe, M. (2010). Interior point and other algorithms for solving the Hamiltonian cycle problem. , University of South Australia.Google Scholar
[15] Langville, A. N. and Meyer, C. D. (2006). Google's PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press, Princeton, NY.Google Scholar
[16] Litvak, N. and Ejov, V. (2009). Markov chains and optimality of the Hamiltonian cycle. Math. Operat. Res. 34, 71–82.Google Scholar
[17] Seneta, E. (1981). Nonnegative Matrices and Markov Chains, 2nd edn. Springer, New York.Google Scholar
[18] Serra-Capizzano, S. (2005). Jordan canonical form of the Google matrix: a potential contribution to the PageRank computation. SIAM J. Matrix Anal. Appl. 27, 305–312.Google Scholar
[19] Volkovich, Y. V. (2009). Stochastic analysis of web page ranking. , University of Twente.Google Scholar