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

  • Vladimir Ejov (a1), Nelly Litvak (a2), Giang T. Nguyen (a1) and Peter G. Taylor (a3)

Abstract

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.

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Copyright

Corresponding author

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

References

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[2] Borkar, V. S., Ejov, V. and Filar, J. A. (2004). Directed graphs, Hamiltonicity and doubly stochastic matrices. Random Structures Algorithms 25, 376395.
[3] Borkar, V. S., Ejov, V. and Filar, J. A. (2009). On the Hamiltonicity gap and doubly stochastic matrices. Random Structures Algorithms 34, 502519.
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Keywords

MSC classification

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

  • Vladimir Ejov (a1), Nelly Litvak (a2), Giang T. Nguyen (a1) and Peter G. Taylor (a3)

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