In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph (whether directed or undirected). Both problems are NP-complete. The problem of finding a Hamiltonian cycle or path is in FNP.
There is a simple relation between the two problems. The Hamiltonian path problem for graph G is equivalent to the Hamiltonian cycle problem in a graph H obtained from G by adding a new vertex and connecting it to all vertices of G.
The Hamiltonian cycle problem is a special case of the traveling salesman problem, obtained by setting the distance between two cities to a finite constant if they are adjacent and infinity otherwise.
The directed and undirected Hamiltonian cycle problems were two of Karp's 21 NP-complete problems. Garey and Johnson showed shortly afterwards in 1974 that the directed Hamiltonian cycle problem remains NP-complete for planar graphs and the undirected Hamiltonian cycle problem remains NP-complete for cubic planar graphs.
A randomized algorithm for Hamiltonian path that is fast on most graphs is the following: Start from a random vertex, and continue if there is a neighbor not visited. If there are no more unvisited neighbors, and the path formed isn't Hamiltonian, pick a neighbor uniformly at random, and rotate using that neighbor as a pivot. (That is, add an edge to that neighbor, and remove one of the existing edges from that neighbor so as not to form a loop.) Then, continue the algorithm at the new end of the path.
Due to the complexity of the problem computers have to be used to solve what may seem to be minor tasks, for example to calculate the shortest route to tour the ten biggest cities in the UK over 3.5 million routes have to be analysed.
In July 2009, research published in the Journal of Biological Engineering showed that a bacterial computer can be used to solve a simple Hamiltonian path problem (using three locations).