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Topic: Hamiltonian path problem

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	Hamiltonian path problem - Wikipedia, the free encyclopedia
		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).
		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 unity if they are adjacent and infinity otherwise.
	en.wikipedia.org /wiki/Hamiltonian_path_problem (291 words)

	ipedia.com: Hamiltonian path Article (Site not responding. Last check: 2007-09-23)
		In graph theory, a Hamiltonian path is a path that visits each vertex exactly once.
		A Hamiltonian cycle (also called Hamiltonian circuit, vertex tour or graph cycle) is a cycle that visits each vertex exactly once, except for the starting vertex.
		Similar notions may be defined for directed graphss, where edges (arcs) of a path or a cycle are required to point in the same direction, i.e., connected tail-to-head.
	www.ipedia.com /hamiltonian_path.html (500 words)

	Hamiltonian path problem -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-23)
		Both problems are (Click link for more info and facts about NP-complete) NP-complete.
		The problem of finding a Hamiltonian cycle or path is in (Click link for more info and facts about FNP) FNP.
		The Hamiltonian cycle problem is a special case of the (Click link for more info and facts about traveling salesman problem) traveling salesman problem, obtained by setting the distance between two cities to unity if they are adjacent and infinity otherwise.
	www.absoluteastronomy.com /encyclopedia/h/ha/hamiltonian_path_problem.htm (179 words)

	Molecular Computation of Solutions to Combinatorial Problems
		During the separation step, molecules encoding Hamiltonian paths may fail to bind adequately and be lost, whereas molecules encoding non-Hamiltonian paths may bind nonspecifically and be retained.
		Nonetheless, for certain intrinsically complex problems, such as the directed Hamiltonian path problem where existing electronic computers are very inefficient and where massively parallel searches can be organized to take advantage of the operations that molecular biology currently provides, it is conceivable that molecular computation might compete with electronic computation in the near term.
		It is a research problem of considerable interest to elucidate the kinds of algorithms that are possible with the use of molecular methods and the kinds of problems that these algorithms can efficiently solve (12, 15, 16).
	www.apl.jhu.edu /Notes/Boon/Common/DNA_computing.htm (3289 words)

	Leonard Adleman (Site not responding. Last check: 2007-09-23)
		This problem is difficult for conventional computers to solve because it is a "non-deterministic polynomial time problem".
		The Hamiltonian Path problem was chosen by Adleman because it is known problem.
		After generating the numerous random paths in the first step, he used polymerase chain reaction (PCR) to amplify and keep only the paths that began on vertex 1 and ended at vertex 6.
	php.iupui.edu /~pellison/n301/page4.html (406 words)

	distributed
		A problem that seems ideally suited for this demonstration is the directed Hamiltonian path problem (a restricted form of the traveling salesperson problem), one of the class of combinatorially hard problems.
		If a problem is split across several processors, either on the same computer, or on different machines distributed over a network, which communicate either via message passing or through a shared memory mechanism, it can be solved in much less time than by performing the computation on a single processor.
		A Hamiltonian path exists in a graph only if there is a connected path from a designated starting node to a designated output node that visits each node of the graph exactly once.
	www.isi.edu /~lerman/etc/distributed.html (1740 words)

	Knight's Tour Notes, Part Cc: Chronology 1900 to Present
		Problem 736 (WJ), 985-986 (TRD), 987-991 (DHH), 1061 (TRD), 1062-63 (WJ), 1064 (HP).
		Problems 1132-35 (FD), 1303-06, 1449-52, 1525-28, 1593-96 (all TRD).
		Problems 1674 -77 (TRD), 1704 (SHH, squares in a circle), 1705 (TRD, squares in a circle), 1813-1816, 1834-37, 1917-20 (all TRD).
	www.ktn.freeuk.com /cc.htm (7794 words)

	Chess Guide > Knight Tour
		The Knight's Tour is a mathematical problem involving a knight on a chessboard.
		There are several billion solutions to the problem, of which about 122,000,000 have the knight finishing on the same square on which it begins.
		The knight's tour problem is an instance of the more general hamiltonian path problem in graph theory, which is NP-complete.
	www.chess.freegames.eu.com /problems-puzzles/knight_tour (260 words)

	Islamset - Computing with DNA: Rediscovering Biology ,Hamiltonian Path Problem, Seven Days in a Lab, A New Field ... (Site not responding. Last check: 2007-09-23)
		The problem I chose was the Hamiltonian Path Problem.
		Your job (the Hamiltonian Path Problem) is to determine if a sequence of connecting flights (a path) exists that starts in Atlanta (the start vertex) and ends in Detroit (the end vertex), while passing through each of the remaining cities (Boston and Chicago) exactly once.
		The Hamiltonian Path Problem is to decide for any given graph with specified start and end vertices whether a Hamiltonian path exists or not.
	www.islamset.com /healnews/dna/dna.html (4075 words)

	Hamiltonian Cycle
		Discussion: The problem of finding a Hamiltonian cycle or path in a graph is a special case of the traveling salesman problem, one where each pair of vertices with an edge between them is considered to have distance 1, while nonedge vertex pairs are separated by distance
		Closely related is the problem of finding the longest path or cycle in a graph, which occasionally arises in pattern recognition problems.
		We seek the longest path in a graph where the weight of an edge (x,y) is the number of points x beat y by.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE176.HTM (779 words)

	dna-abc3 (Site not responding. Last check: 2007-09-23)
		That is why so much excitement greeted Leonard Adleman's announcement (``Molecular computation of solutions to combinatorial problems,'' Science, 266 1021-1024, November 11, 1994) that he had found a way to solve the path problem using individual molecules as computing elements.
		This was like having six hundred million million (albeit very rudimentary) computers working on the problem at the same time: the age-old parable of the monkeys and the typewriters brought to life in a test-tube.
		The path problem can be parallelized as follows: imagine a huge number of passengers traveling the airline and making allowable connections at random.
	80-www.ams.org.library.uor.edu /featurecolumn/archive/dna-abc3.html (506 words)

	Yuan Xie
		The Hamiltonian Path Problem is a member of a class of computational problems called NP-Complete Problems, which are a group of problems that have been proven to be among the most difficult problems that Computer Scientists have attempted to solve.
		The goal of the Directed Hamiltonian Path Problem is to decide if a path exists in a given directed graph that visits all of the vertices in the graph exactly once.
		After all of the possible paths – of all lengths – have been generated through the self assembly of the designed DNA sequences, the second step is to sort the vertices (Winfree et al, 1996).
	www.cco.caltech.edu /~sciwrite/journal03/xie.html (3282 words)

	RSA Security - 2.3.12 What are some other hard problems?
		A few examples of hard problems are the Traveling Salesman Problem, the Integer and Mixed Integer Programming Problem, the Graph Coloring Problem, the Hamiltonian Path Problem and the Satisfiability Problem for Boolean Expressions.
		The Graph Coloring Problem is to determine whether a graph can be colored with a fixed set of colors such that no two adjacent vertices have the same color, and to produce such a coloring.
		Another hard problem is the Knapsack Problem, a narrow case of the Subset Sum Problem (see Question 2.3.11).
	www.rsasecurity.com /rsalabs/node.asp?id=2198 (324 words)

	DNA Computing
		However, all known algorithms for this problem have exponential worst-case complexity and hence there are instances of modest size for which these algorithms require an impractical amount of computer time to render a decision.
		Since the directed Hamiltonian path problem has been proven to be NP-complete, it seem likely that no efficient (that is, polynomial time) algorithm exists for solving it.
		There are lots of problems which come from the graph theory or from operations research that are also hard solvable problems and are worth solving with DNA computing.
	www.casi.net /D.BioInformatics1/D.Fall2000ClassPage/DC1/dc.htm (2738 words)

	DNA Computing Examples (Site not responding. Last check: 2007-09-23)
		The problem is a Hamiltonian Path problem - a problem involving paths going though points using certain rules and a general rule that no point can be passed through more than once.
		The particular Hamiltonian Path problem that Adleman solved is popularly referred to as a "traveling salesman" problem.
		In our problem, we will have five cities, Cincinnati, Columbus, Cleveland, Dayton, and Toledo that are connected as follows: Cincinnati is connected to Dayton and Columbus, Dayton is connected to Toledo and Columbus, Columbus is connected to Toledo and Cleveland, and Cleveland is connected to Columbus and Toledo.
	www.ih.k12.oh.us /esdiscovery5/genetics/examplesDNA.htm (349 words)

	[No title] (Site not responding. Last check: 2007-09-23)
		A Hamiltonian path is a simple open path that contains each vertex in a graph exactly once.
		The Hamiltonian Path problem is the problem to determine whether a given graph contains a Hamiltonian path.
		Call the problem to determine if these jobs can be executed in 3 time units Schedule and show that it is NP-complete.
	www.nada.kth.se /theory/ads/94-95/oevning9 (133 words)

	dna-abc2 (Site not responding. Last check: 2007-09-23)
		In these terms, the directed Hamiltonian path problem for this graph is to find a sequence of flights beginning in Fresno, ending in Boston and visiting each airport in the route map exactly once.
		This problem is simple to state and, when there are only few airports and flights, can be solved by inspection.
		But the amount of computation necessary to settle the question, using the most efficient algorithms known at present, grows exponentially with the size of the route map: essentially the only way is to go down the list of all possible chains of flights until a solution is found or until the list is exhausted.
	80-www.ams.org.library.uor.edu /featurecolumn/archive/dna-abc2.html (275 words)

	DNA Computing
		This problem is a member of the large class of notoriously intractable NP-complete problems, as are the Travelling Salesman and Bin Packing problems.
		These problems are characterised by an exponential- size search space; a problem of size 10 may take a fraction of a second to solve on a PC, but one of size 30 may take years.
		Although this particular problem could be solved on a piece of paper in under an hour, when the number of cities is increased to 70, the problem becomes too complex for even a supercomputer.
	www.casi.net /D.BioInformatics1/D.Fall2000ClassPage/DC2/dc2.htm (2548 words)

	Graph Definitions
		The Edge Connectivity of an Undirected Graph is the minimum number of edges that must be removed to "disconnect the graph." This is a number.
		A value of zero means the given graph has at least one pair of vertices x, y in V such that there is no path connecting x and y.
		Bipartite - a Graph with a set of vertices that can be divided into exactly two non empty subsets such that no edge connects two vertices within a subset and every vertex in one subset has at least one Edge to a vertex in the other subset.
	www.csee.umbc.edu /~squire/reference/graph_def.shtml (820 words)

	JYI Volume Eight Features: Fear Not Traveling Salesmen, DNA Computing is Here to Save the Day (Site not responding. Last check: 2007-09-23)
		In a Hamiltonian Path problem, a series of towns are connected to each other by a fixed number of bridges.
		As the number of cities grows, the problem generates too many possible paths for brute force solving, so a computer is needed to solve it.
		Although the solution to Adleman’s seven-city Hamiltonian Path problem was relatively straightforward (since all possible routes can be written by hand in a reasonable amount of time), his experiment showed that DNA could be useful as a computational tool.
	www.jyi.org /volumes/volume8/issue2/features/srivastava.html (2301 words)

	[No title] (Site not responding. Last check: 2007-09-23)
		The Hamiltonian Path Problem The Hamiltonian path problem (hereafter referred to as HP) in a directed graph with designated start and end vertices is a very hard problem to solve for a general directed graph, it is NP-C, so there are no polynomial time solutions to this problem.
		Notice one thing about problems in NP (and NP-C), if we are given an input for the problem and a proposed solution, it is possible to check the validity of the solution in polynomial time.
		Note that being able to check one path in polynomially bounded time does not give us an algorithm for solving the HP problem because the number of distinct paths to be checked is not, in general, polynomially bounded.
	longwood.cs.ucf.edu /courses/cop3530-su02/fut19.doc (7111 words)

	COMPUTING WITH DNA
		The directed Hamiltonian path problem would be: Could a
		directed Hamiltonian path problem is one of a class of problems for
		problem from all of the others that formed in the test tube.
	www.usc.edu /uscnews/stories/935.html (931 words)

	Wired 3.08: Gene Genie
		A Hamiltonian path problem involving four or five cities can be solved by doodling on a piece of paper, but when the number of cities grows by even a small amount, the problem's difficulty balloons - it becomes what is known in mathematical terms as "hard." Hard problems cannot be solved efficiently by algebraic equations.
		Finding a Hamiltonian path connecting 100 cities using a well-known algorithm, for example, would take 10147 operations.
		He was looking for the one itinerary known to connect the cities in a directed Hamiltonian path - a path that would begin in Atlanta, end in Detroit, and pass through each intervening city only once.
	www.wired.com /wired/archive/3.08/molecular.html (915 words)

	A Heuristic Approach for Hamiltonian Path Problem with Molecules - Arita, Suyama, Hagiya (ResearchIndex) (Site not responding. Last check: 2007-09-23)
		In this method, longer paths are generated by combining shorter paths and by eliminating paths containing duplicate vertices.
		Reflective PCR is used to discriminate paths with duplicate vertices from those without...
		A heuristic approach for Hamiltonian Path Problem with molecules.
	citeseer.ist.psu.edu /arita97heuristic.html (503 words)

	CSCI 3104 Class notes Page 20 (Site not responding. Last check: 2007-09-23)
		The problem to be solved is the "Hamiltonian Path" problem...
		The version of the problem considered is for directed graphs.
		The problem is NP-hard (the article only says "hard", but we know better).
	www.cs.colorado.edu /~karl/3104.fall95/20.html (234 words)

	Abstract (Site not responding. Last check: 2007-09-23)
		In 1959, Richard Feynman gave a visionary talk describing the possibility of building computers that were "sub-microscopic." Last fall, in a revolutionary paper whose consequences may prove to be among the major scientific breakthroughs of this century, Leonard Adleman showed how computations could be carried out using simple DNA manipulations.
		I will give an overview of how computational problems are ranked by difficulty, leading to the important class of "NP-complete" problem, and why the massive parallelism of solution-phase chemistry was crucial to Adleman's work.
		I will then give a ("nondeterministic") algorithm for solving the directed Hamiltonian path problem and outline how the inputs were encoded and the algorithm carried out by Adleman at the molecular level, where a complete solution was encoded in a single molecule.
	www.geom.uiuc.edu /~barzilai/cr/dna-dir/dna.abstract.spring95.html (166 words)

	1.5.5 Hamiltonian Cycle (Site not responding. Last check: 2007-09-23)
		Problem: Find an ordering of the vertices such that each vertex is visited exactly once.
		Excerpt from The Algorithm Design Manual: The problem of finding a Hamiltonian cycle or path in a graph is a special case of the traveling salesman problem, one where each pair of vertices with an edge between them has distance 1, while nonedge vertex pairs are separated by distance infinity.
		The longest path through this graph is likely the correct interpretation.
	www.cs.sunysb.edu /~algorith/files/hamiltonian-cycle.shtml (195 words)

	CSCI 3104 Class notes Page 21 (Site not responding. Last check: 2007-09-23)
		In the end, this method is exhaustive search (with the terrible asymptotic performance intrinsic to exhaustive search) with an impressive constant factor.
		That's a very small number compared to the number of paths of length 100 in a graph where each node is connected to 10 other nodes.
		It also may provide a way to increase the size of Hamiltonian Path problems (and other NP-complete problems) that we can handle.
	www.cs.colorado.edu /~karl/3104.fall95/21.html (180 words)

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