Empirical Study of Phase Transition of Hamiltonian Cycle Problem in Random Graphs with Degrees Greater Than One

摘要

It is well-known that many NP-complete problems will undergo phase transitions along with the change of some problem-specific critical parameter values. It has been shown that the phase transition will occur at an average node degree log(n) + log log(n) for Hamiltonian cycle problem in random graphs with n nodes. In this paper, we prove that random graphs with such critical average node degrees tend to be hamiltonian graphs if their node degrees are greater than one. Using an improved backtracking algorithm with pruning operations, we try to find the areas where hard problem instances can be found with high probability. For random graphs with degrees greater than 1, the experimental results have demonstrated that hard cases can be found with high probability when graphs take lower average degrees, and the phase transition occurs at lower average degrees, too. Empirically, the phase transition between hamiltonicity and non-hamiltonicity occurs when the average degree is 1.1601+0.2418log(n) for random graphs with degrees greater than one.