Hamiltonian path problem

The problems of finding a Hamiltonian path and a Hamiltonian cycle can be related as follows:

• In one direction, the Hamiltonian path problem for graph G can be related to the Hamiltonian cycle problem in a graph H obtained from G by adding a new universal vertex x, connecting x to all vertices of G. Thus, finding a Hamiltonian path cannot be significantly slower (in the worst case, as a function of the number of vertices) than finding a Hamiltonian cycle.
• In the other direction, the Hamiltonian cycle problem for a graph G is equivalent to the Hamiltonian path problem in the graph H obtained by adding terminal (degree-one) vertices s and t attached respectively to a vertex v of G and to v', a cleaved copy of v which gives v' the same neighbourhood as v. The Hamiltonian path in H running through vertices ${\displaystyle s-v-x-\cdots -y-v'-t}$ corresponds to the Hamiltonian cycle in G running through ${\displaystyle v-x-\cdots -y(-v)}$.[4]

To decide if a graph has a Hamiltonian path, one would have to check each possible path in the input graph G. There are n! different sequences of vertices that might be Hamiltonian paths in a given n-vertex graph (and are, in a complete graph), so a brute force search algorithm that tests all possible sequences would be very slow.

An early exact algorithm for finding a Hamiltonian cycle on a directed graph was the enumerative algorithm of Martello.[3] A search procedure by Frank Rubin[5] divides the edges of the graph into three classes: those that must be in the path, those that cannot be in the path, and undecided. As the search proceeds, a set of decision rules classifies the undecided edges, and determines whether to halt or continue the search. The algorithm divides the graph into components that can be solved separately.

Also, a dynamic programming algorithm of Bellman, Held, and Karp can be used to solve the problem in time O(n² 2ⁿ). In this method, one determines, for each set S of vertices and each vertex v in S, whether there is a path that covers exactly the vertices in S and ends at v. For each choice of S and v, a path exists for (S,v) if and only if v has a neighbor w such that a path exists for (S − v,w), which can be looked up from already-computed information in the dynamic program.[6][7]

Andreas Björklund provided an alternative approach using the inclusion–exclusion principle to reduce the problem of counting the number of Hamiltonian cycles to a simpler counting problem, of counting cycle covers, which can be solved by computing certain matrix determinants. Using this method, he showed how to solve the Hamiltonian cycle problem in arbitrary n-vertex graphs by a Monte Carlo algorithm in time O(1.657ⁿ); for bipartite graphs this algorithm can be further improved to time O(1.415ⁿ).[8]

For graphs of maximum degree three, a careful backtracking search can find a Hamiltonian cycle (if one exists) in time O(1.251ⁿ).[9]

Hamiltonian paths can be found using a SAT solver. The Hamiltonian path is NP-Complete meaning it can be mapping reduced to the 3-SAT problem. As a result, finding a solution to the Hamiltonian Path problem is equivalent to finding a solution for 3-SAT.

Because of the difficulty of solving the Hamiltonian path and cycle problems on conventional computers, they have also been studied in unconventional models of computing. For instance, Leonard Adleman showed that the Hamiltonian path problem may be solved using a DNA computer. Exploiting the parallelism inherent in chemical reactions, the problem may be solved using a number of chemical reaction steps linear in the number of vertices of the graph; however, it requires a factorial number of DNA molecules to participate in the reaction.[10]

An optical solution to the Hamiltonian problem has been proposed as well.[11] The idea is to create a graph-like structure made from optical cables and beam splitters which are traversed by light in order to construct a solution for the problem. The weak point of this approach is the required amount of energy which is exponential in the number of nodes.

Networks on Chip are used in computer systems and processors serving as communication for on-chip components.[24] The performance of NoC is determined by the method it uses to move packets of data across a network.[25] The Hamiltonian Path problem can be implemented as a path-based method in multicast routing. Path-based multicast algorithms will determine if there is a Hamiltonian path from the start node to each end node and send packets across the corresponding path. Utilizing this strategy guarantees deadlock and livelock free routing, increasing the efficiency of the NoC.[26]

Rendering engines are a form of software used in computer graphics to generate images or models from input data.[27] In three dimensional graphics rendering, a common input to the engine is a polygon mesh. The time it takes to render the object is dependent on the rate at which the input is received, meaning the larger the input the longer the rendering time. For triangle meshes, however, the rendering time can be decreased by up to a factor if three. This is done through "ordering the triangles so that consecutive triangles share a face."[28] That way, only one vertex changes between each consecutive triangle. This ordering exists if the dual graph of the triangular mesh contains a Hamiltonian path.