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The Hamiltonian cycle problem (HCP) is an important combinatorial problemwith applications in many areas. It is among the first problems used forstudying intrinsic properties, including phase transitions, of combinatorialproblems. While thorough theoretical and experimental analyses have been madeon the HCP in undirected graphs, a limited amount of work has been done for theHCP in directed graphs (DHCP). The main contribution of this work is aneffective algorithm for the DHCP. Our algorithm explores and exploits the closerelationship between the DHCP and the Assignment Problem (AP) and utilizes atechnique based on Boolean satisfiability (SAT). By combining effectivealgorithms for the AP and SAT, our algorithm significantly outperforms previousexact DHCP algorithms, including an algorithm based on the award-winningConcorde TSP algorithm. The second result of the current study is anexperimental analysis of phase transitions of the DHCP, verifying and refininga known phase transition of the DHCP.

An Effective Algorithm for and Phase Transitions of the Directed Hamiltonian Cycle Problem