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In this paper, we present some algebraic properties of a particular class of probability transition matrices, namely, Hamiltonian transition matrices. Each matrix $$P$$ in this class corresponds to a Hamiltonian cycle in a given graph $$G$$ on $$n$$ nodes and to an irreducible, periodic, Markov chain. We show that a number of important matrices traditionally associated with Markov chains, namely, the stationary, fundamental, deviation and the hitting time matrix all have elegant expansions in the first $$n-1$$ powers of $$P$$ , whose coefficients can be explicitly derived. We also consider the resolvent-like matrices associated with any given Hamiltonian cycle and its reverse cycle and prove an identity about the product of these matrices. As an illustration of these analytical results, we exploit them to develop a new heuristic algorithm to determine a non-Hamiltonicity of a given graph. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new hybrid algorithm for the Hamiltonian cycle problem by synthesizing the Cross Entropy method and Markov decision processes. In particular, this new algorithm assigns a random length to each arc and alters the Hamiltonian cycle problem to the travelling salesman problem. Thus, there is now a probability corresponding to each arc that denotes the probability of the event 'this arc is located on the shortest tour.' Those probabilities are then updated as in cross entropy method and used to set a suitable linear programming model. If the solution of the latter yields any tour, the graph is Hamiltonian. Numerical results reveal that when the size of graph is small, say less than 50 nodes, there is a high chance the algorithm will be terminated in its cross entropy component by simply generating a Hamiltonian cycle, randomly. However, for larger graphs, in most of the tests the algorithm terminated in its optimization component (by solving the proposed linear program). [ABSTRACT FROM AUTHOR]

We develop a new, random walk-based, algorithm for the Hamiltonian cycle problem. The random walk is on pairs of extreme points of two suitably constructed polytopes. The latter are derived from geometric properties of the space of discounted occupational measures corresponding to a given graph. The algorithm searches for a measure that corresponds to a common extreme point in these two specially constructed polyhedral subsets of that space. We prove that if a given undirected graph is Hamiltonian, then with probability one this random walk algorithm detects its Hamiltonicity in a finite number of iterations. We support these theoretical results by numerical experiments that demonstrate a surprisingly slow growth in the required number of iterations with the size of the given graph. [ABSTRACT FROM AUTHOR]