By Jerzy A. Filar
Managed Markov Chains, Graphs & Hamiltonicity summarizes a line of analysis that maps definite classical difficulties of discrete arithmetic - akin to the Hamiltonian cycle and the touring Salesman difficulties - into convex domain names the place continuum research may be conducted.
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Extra resources for Controlled Markov chains, graphs, and Hamiltonicity
Let fh be the standard HC on four nodes: 1 → 2 → 3 → 4 → 1. This Hamiltonian Cycle induces the perturbed probability transition matrix Pε (fh ) of the form: ε 1 − 3ε ε ε ε 1 − 3ε ε ε Pε (fh ) = . ε ε 1 − 3 ε ε 1 − 3ε ε ε ε Since Pε∗ (fh ) = 1/4J, the matrix needed to calculate the fundamental matrix induced by fh is 5/4 − ε −ε + 1/4 1 I − Pε (fh ) + J = −ε + 1/4 4 −3/4 + 3 ε −3/4 + 3 ε 5/4 − ε −ε + 1/4 −ε + 1/4 −ε + 1/4 −3/4 + 3 ε 5/4 − ε −ε + 1/4 −ε + 1/4 −ε + 1/4 .
However, it is a relatively crude estimate that applies to all graphs. Better lower bounds — applying to specific classes of graphs — might form measures of the difficulty of “NP-hardness” of the HCP in these cases. ” Such notions are yet to be rigorously formalized. 2 Determinant of the Inverse of Fundamental Matrix In this section, we describe some interesting results that, in principle, could be considered as purely algebraic properties of certain matrices and their eigenvalues and determinants.
However, if the policy f is non-Hamiltonian, it is “likely” to be either tracing out short cycles (of length less than N ) before returning to node 1, or getting trapped for a long time in some ergodic class not containing node 1. Both of these effects would increase the variability of τ1 . Of course, if the interpretation of the HCP as a variance minimization problem is to be useful a formula for evaluating this variance is needed. The next result (see also ) shows that, for a doubly stochastic policy, such a variance is closely related to the top left hand corner element of the fundamental matrix induced by that policy.
Controlled Markov chains, graphs, and Hamiltonicity by Jerzy A. Filar
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