Markov chains and optimality of the Hamiltonian cycle

We consider the Hamiltonian cycle problem (HCP) embedded in a controlled Markov decision process. In this setting, HCP reduces to an optimization problem on a set of Markov chains corresponding to a given graph. We prove that Hamiltonian cycles are minimizers for the trace of the fundamental matrix on a set of all stochastic transition matrices. In case of doubly stochastic matrices with symmetric linear perturbation, we show that Hamiltonian cycles minimize a diagonal element of a fundamental matrix for all admissible values of the perturbation parameter. In contrast to the previous work on this topic, our arguments are primarily based on probabilistic rather than algebraic methods

Improved upper bounds on even-cycle creating Hamilton paths

We study the function $H_n(C_{2k})$, the maximum number of Hamilton paths such that the union of any pair of them contains $C_{2k}$ as a subgraph. We give upper bounds on this quantity for $k\in \{3, 4, 5\}$, improving results of Harcos and Solt\'esz, and we show that if a conjecture of Ustimenko is true then one additionally obtains improved upper bounds for all $k\geq 6$. In order to prove our results, we extend a theorem of Krivelevich which counts Hamilton cycles in $(n, d, \lambda)$-graphs to graphs which are not regular, and then apply this result to graphs constructed from polarity graphs of generalized polygons

On transition matrices of Markov chains corresponding to Hamiltonian cycles

International audienceIn 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

On the fastest finite Markov processes

International audienceConsider a finite irreducible Markov chain with invariant probability π. Define its inverse communication speed as the expectation to go from x to y, when x, y are sampled independently according to π. In the discrete time setting and when π is the uniform distribution υ, Litvak and Ejov have shown that the permutation matrices associated to Hamiltonian cycles are the fastest Markov chains. Here we prove (A) that the above optimality is with respect to all processes compatible with a fixed graph of permitted transitions (assuming that it does contain a Hamiltonian cycle), not only the Markov chains, and, (B) that this result admits a natural extension in both discrete and continuous time when π is close to υ: the fastest Markov chains/processes are those moving successively on the points of a Hamiltonian cycle, with transition probabilities/jump rates dictated by π. Nevertheless, the claim is no longer true when π is significantly different from υ

Advances in Branch-and-Fix methods to solve the Hamiltonian cycle problem in manufacturing optimization

159 p.Esta tesis parte del problema de la optimización de la ruta de la herramienta donde se contribuye con unsistema de soporte para la toma de decisiones que genera rutas óptimas en la tecnología de FabricaciónAditiva. Esta contribución sirve como punto de partida o inspiración para analizar el problema del cicloHamiltoniano (HCP). El HCP consiste en visitar todos los vértices de un grafo dado una única vez odeterminar que dicho ciclo no existe. Muchos de los métodos propuestos en la literatura sirven paragrafos no dirigidos y los que se enfocan en los grafos dirigidos no han sido implementados ni testeados.Uno de los métodos para resolver el problema es el Branch-and-Fix (BF), un método exacto que utiliza latranformación del HCP a un problema continuo. El BF es un algoritmo de ramificación que consiste enconstruir un árbol de decisión donde en cada vértice dos problemas lineales son resueltos. Este método hasido testeado en grafos de tamaño pequeño y por ello, no se ha estudiado en profundidad las limitacionesque puede presentar. Por ello, en esta tesis se proponen cuatro contribuciones metodológicasrelacionadas con el HCP y el BF: 1) mejorar la enficiencia del BF en diferentes aspectos, 2) proponer unmétodo de ramificación global, 3) proponer un método del BF colapsado, 4) extender el HCP a unescenario multi-objetivo y proponer un método para resolverlo