A hybrid simulation-optimization algorithm for the Hamiltonian cycle problem

Abstract

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).

Hamiltonian Cycles and Subsets of Discounted Occupational Measures

“…Eshragh et al [13] transformed the polytope F β (G) by changing variables x ij := (1−β n )y ij for all (i, j) ∈ E to produce the polytope H β (G) ⊆ R |E| defined by the constraints…”

“…Let A B denote the (n + 1) × (n + 1)-submatrix of A corresponding to B. If A B is invertible and x is the vector whose coordinates corresponding to B are given by A −1 B b while all other coordinates are zero, then it has been proved in [13,Lemma 3.2] that x k k+1 = β k−1 > 0 for k ∈ {1, 2, . .…”

“…In order to fully investigate the efficiency of the wedge constraints, we need to determine the structure and prevalence of feasible bases of these new extreme points of the polytope WH β (G). Analogous to Eshragh et al [13], we introduce the concept of quasi-Hamiltonian bases for WH β (G) as follows.…”

“…Feinberg showed that if the graph G is Hamiltonian, the polytope F β (G) has an extreme point, called a Hamiltonian extreme point, for each of its Hamiltonian cycles. Subsequently, Ejov et al [10] described some geometric properties of F β (G) and Eshragh et al [13] transformed F β (G) to a polytope H β (G) to improve algorithmic efficiency. In 2011, Eshragh and Filar [12] partitioned all extreme points of H β (G) into five types, consisting of Hamiltonian extreme points and non-Hamiltonian extreme points of types 1, 2, 3 and 4.…”

An improved incremental least-mean-squares algorithm for distributed estimation over wireless sensor networks

“…Distributed consensus-based Kalman filtering algorithm that combines consensus strategy into the standard Kalman filter has been well used for sensor schedule 14 and localization of target. 15 Originating from the consensus strategy, the incremental adaptive strategy requires a ring topology like the Hamiltonian cycle, 16,17 where each node is visited exactly once per iteration. So that the local measurements and the results from previous node on cycle path are fused to generate the new estimate of current node by iteration algorithm (e.g.…”