A practical efficient and effective method for the Hamiltonian cycle problem that runs on a standard computer

Given $N$ cities and $R < N² - N$ directed (unidirectional/one way) roads does there exist a tour of all $N$ cities stopping at each city exactly once using the given roads (a Hamiltonian cycle)? This Hamiltonian cycle problem (HCP) is an NP-complete problem, for which there is no known polynomial time solution algorithm. The HCP has important practical applications, for example, to logistical problems. It was claimed that an adiabatic quantum computer could solve an NP-complete problem faster than classical algorithms, but claim appears to have been debunked. Here we demonstrate an algorithm which runs on a standard computer that efficiently and effectively solves the HCP for at least up to 500 cities: We first optimized a simulated annealing based algorithm used for smaller sized HCP problems. Then we found that when a tour was deliberately inserted in a list of otherwise randomly chosen roads, crucially, if "extra" random roads are added to bring the total number of roads up to $0.58 N loge N$ or more there is a 100% chance our algorithm will find a HC, but conversely when a list of roads does not include a pre-inserted tour random roads have to be added until there are $0.9 N loge N$ roads to have a chance of finding a HC. We found similarly for a set of roads non-randomly chosen. Thus, the presence of a HC in a set of roads induces "connectivity" throughout the roads and a HC can be found with an insertion of a modest number of extra roads. Our algorithm also shows that only weakly non-local information is needed to find an HCP that is a global state.