Article

# Determinants and Longest Cycles of Graphs.

SIAM Journal on Discrete Mathematics (Impact Factor: 0.58). 01/2008; 22:1215-1225. DOI: 10.1137/070693898

Source: DBLP

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**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.Journal of Applied Probability 01/2011; 48(2011). · 0.69 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We show that the determinant objective function introduced in Ejov et al. [V. Ejov, J. A. Filar, W. Murray, G.T. Nguyen, Determinants and longest cycles of graph, SIAM J. Discrete Math. 22 (33) (2008) 1215–1225] performs well under a certain symmetric linear perturbation. That means sub-graphs corresponding to Hamiltonian cycles of a given graph are maximizers over the hull of all sub-graphs with perturbation parameter ε∈[0,1). Note that in other optimization formulations (see, for example [V.S. Borkar, V. Ejov, J.A. Filar, Directed graphs, Hamiltonicity and doubly stochastic matrices, Random Structures Algorithms 25 (2004) 376–395; V. Ejov, J.A. Filar, M. Nguyen, Hamiltonian cycles and singularly perturbed Markov chains, Math. Oper. Res. 29 (1) (2004) 114–131; J.A. Filar, K. Liu, Hamiltonian cycle problem and singularly perturbed Markov decision process, in: Statistics, Probability and Game Theory: Papers in Honor of David Blackwell, IMS Lecture Notes – Monograph Series, USA, 1996]), ε in the corresponding perturbation was required to be significantly small.Linear Algebra and its Applications 01/2009; 431(5):543-552. · 0.98 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We present an algorithm to find the determinant and its first and second derivatives of a rank-one corrected generator matrix of a doubly stochastic Markov chain. The motivation arises from the fact that the global minimiser of this determinant solves the Hamiltonian cycle problem. It is essential for algorithms that find global minimisers to evaluate both first and second derivatives at every iteration. Potentially the computation of these derivatives could require an overwhelming amount of work since for the Hessian N 2 cofactors are required. We show how the doubly stochastic structure and the properties of the objective may be exploited to calculate all cofactors from a single LU decomposition.Journal of Global Optimization 08/2013; 56(4). · 1.36 Impact Factor

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