Hamiltonian path problem and AlgaePARC

This article is about the specific problem of determining whether a Hamiltonian path or cycle exists in a given graph. For the general graph theory concepts, see Hamiltonian path.

In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path (a path in an undirected or directed graph that visits each vertex exactly once) or a Hamiltonian cycle exists in a given graph (whether directed or undirected). Both problems are NP-complete.

There is a simple relation between the problems of finding a Hamiltonian path and a Hamiltonian cycle. In one direction, 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. 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, a graph G has a Hamiltonian cycle using edge uv if and only if the graph H obtained from G by replacing the edge by a pair of vertices of degree 1, one connected to u and one connected to v, has a Hamiltonian path. Therefore, by trying this replacement for all edges incident to some chosen vertex of G, the Hamiltonian cycle problem can be solved by at most n Hamiltonian path computations, where n is the number of vertices in the graph. The Hamiltonian cycle problem is also a special case of the travelling salesman problem, obtained by setting the distance between two cities to one if they are adjacent and two otherwise, and verifying that the total distance travelled is equal to n (if so, the route is a Hamiltonian circuit; if there is no Hamiltonian circuit then the shortest route will be longer). Algorithms

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. There are several faster approaches. A search procedure by Frank Rubin 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.

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(1.414n).

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

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. Complexity

The problem of finding a Hamiltonian cycle or path is in FNP; the analogous decision problem is to test whether a Hamiltonian cycle or path exists. The directed and undirected Hamiltonian cycle problems were two of Karp's 21 NP-complete problems. They remain NP-complete even for undirected planar graphs of maximum degree three, for directed planar graphs with indegree and outdegree at most two, for bridgeless undirected planar 3-regular bipartite graphs, and for 3-connected 3-regular bipartite graphs. However, putting all of these conditions together, it remains open whether 3-connected 3-regular bipartite planar graphs must always contain a Hamiltonian cycle, in which case the problem restricted to those graphs could not be NP-complete; see Barnette's conjecture.

In graphs in which all vertices have odd degree, an argument related to the handshaking lemma shows that the number of Hamiltonian cycles through any fixed edge is always even, so if one Hamiltonian cycle is given, then a second one must also exist. However, finding this second cycle does not seem to be an easy computational task. Papadimitriou defined the complexity class PPA to encapsulate problems such as this one.

AlgaePARC and Hamiltonian path problem

Wageningen UR (University & Research centre) is constructing AlgaePARC (Algae Production And Research Centre) at the Wageningen Campus. The goal of AlgaePARC is to fill the gap between fundamental research on algae and full-scale algae production facilities. This will be done by setting up flexible pilot scale facilities to perform applied research and obtain direct practical experience. It is a joined initiative of BioProcess Engineering and Food & Biobased Research of the Wageningen University.

AlgaePARC facility AlgaePARC will start with four different photobioreactors comprising 24 m2 ground surface: an open pond, two types of tubular reactors and a plastic film bioreactor, and a number of smaller systems for the testing of new technologies. This facility is unique, because it is the first facility in which the productivity of four different production systems can be compared during the year under identical conditions. At the same time, knowledge is gained for the development of new photobioreactors and the design of systems on a production scale. For the construction of the facility 2.25 M€ has been made available by the Ministry of Agriculture, Nature and Food Quality (1.5 M€) and the Provincie Gelderland (0.75 M€).

Microalgae Microalgae are currently seen as a promising source of biodiesel and chemical building blocks, which can be used in paint and plastics. Biomass from algae offers a sustainable alternative to products and fuels from the petrochemical industry. This contributes to a biobased economy as algae help to reduce the emissions of carbon dioxide (CO2) and make the economy less dependent on fossil fuels.

AlgaePARC research The costs of biomass produced from algae for biofuels are still ten times too high to be able to compete with today’s other fuels. Within the business community, the question being asked is how it could be produced more cheaply, making it economically viable. Companies within the energy, food, oil and chemical sectors, the Ministry of Agriculture, Nature & Food Quality, the Provincial Government of Gelderland, Oost NV and Wageningen UR are all working together in or contributing to the unique algae research centre AlgaePARC in order to answer that question. See also Algae Microalgae Microbiofuels Photobioreactors Phytoplankton Planktonic algae Biofuel External links AlgaePARC Algae at the WUR
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