Hamiltonian cycle curves in the space of discounted occupational measures
- 145 Downloads
We study the embedding of the Hamiltonian Cycle problem, on a symmetric graph, in a discounted Markov decision process. The embedding allows us to explore the space of occupational measures corresponding to that decision process. In this paper we consider a convex combination of a Hamiltonian cycle and its reverse. We show that this convex combination traces out an interesting “H-curve” in the space of occupational measures. Since such an H-curve always exists in Hamiltonian graphs, its properties may help in differentiating between graphs possessing Hamiltonian cycles and those that do not. Our analysis relies on the fact that the resolvent-like matrix induced by our convex combination can be expanded in terms of finitely many powers of probability transition matrices corresponding to that Hamiltonian cycle. We derive closed form formulae for the coefficients of these powers which are reduced to expressions involving the classical Chebyshev polynomials of the second kind. For regular graphs, we also define a function that is the inner product of points on the H-curve with a suitably defined center of the space of occupational measures and show that, despite the nonlinearity of the inner-product, this function can be expressed as a linear function of auxiliary variables associated with our embedding. These results can be seen as stepping stones towards developing constraints on the space of occupational measures that may help characterize non-Hamiltonian graphs.
KeywordsHamiltonian cycle Markov decision process Occupational measure Chebyshev polynomial Regular graph
This paper is dedicated to Eugene Feinberg on his 60th birthday. We are indebted to M. Haythorpe, K. Clancy, V. Ejov, D. Glynn for their helpful comments and suggestions. This work was supported by the ARC discovery Grant DP 120100532.
- Avrachenkov, K., Eshragh, A., & Filar, J. A. (2014). On transition matrices of Markov chains corresponding to Hamiltonian cycles. Annals of Operations Research. doi: 10.1007/s10479-014-1642-2.
- Christofides, N. (1975). Graph theory: An algorithmic approach. New York: Academic Press.Google Scholar
- Filar, J. A. (2006). Controlled Markov chains, graphs and Hamiltonicity. Foundations and Trends in Stochastic Systems, 1(2), 1–81.Google Scholar
- Filar, J. A., & Lasserre, J.-B. (2000). A non-standard branch and bound method for the Hamiltonian cycle problem. ANZIAM Journal, 42(E), C556–C577.Google Scholar
- Kocay, W., & Li, P.-C. (1994). An algorithm for finding a long path in a graph. Utilitas Mathematica, 45, 169–185.Google Scholar
- Zwillinger, D. (Ed.). (1996). Standard mathematical tables and formulae (30th ed.). Boca Raton, Florida: CRC Press.Google Scholar