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# Topic: Hamiltonian cycle 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)

 Hamiltonian path - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-09) A Hamiltonian cycle is a cycle in an undirected graph which visits each vertex exactly once and also returns to the starting vertex. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once (excluding the start/end vertex). en.wikipedia.org /wiki/Hamiltonian_path   (573 words)

 The Hamiltonian Page Hamiltonian cycles avoiding the arcs of prescribed subtournaments, by J. Bang-Jensen, G. Gutin, and A. Yeo. A sufficient condition for a semicomplete multipartite digraph to be Hamiltonian, by J. Bang-Jensen, G. Gutin and J. Huang. Hamiltonian paths and cycles in hypertournaments, by G. Gutin and A. www.ing.unlp.edu.ar /cetad/mos/Hamilton.html   (1335 words)

 Knight's Tour The Knight's Tour is a mathematical problem involving a knight on a chessboard. The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory, which is NP-complete. The problem of getting a closed knight's tour is similarly an instance of the hamiltonian cycle problem. www.brainyencyclopedia.com /encyclopedia/k/kn/knight_s_tour.html   (282 words)

 PCGrate -> Hamiltonian Cycle Problem It is based on the concept of successive iterative decreases in cycle length, in which the iterative procedure begins with a randomly chosen cycle. As different edges of the graph are assigned the weights 0 and 1, the cycle length is determined in view of the edges with the unit weight. Polynomial complexity of the algorithm results from the fact that the number of iterations for each unit edge of the cycle (i.e., when a correlation is made between the unit edges of the cycle and all zero edges of the graph) is finite. www.pcgrate.com /about/npcomprb/hcp   (688 words)

 On the parallel complexity of the alternating Hamiltonian cycle problem Given a graph with colored edges, a Hamiltonian cycle is called alternating if its successive edges differ in color. The problem of finding such a cycle, even for 2-edge-colored graphs, is trivially NP-complete, while it is known to be polynomial for 2-edge-colored complete graphs. We give a new characterization for such a graph admitting an alternating Hamiltonian cycle which allows us to derive a parallel algorithm for the problem. www.edpsciences.org /articles/ro/abs/1999/04/ro2/ro2.html   (185 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 With a little cleverness, it is sometimes possible to reformulate a Hamiltonian cycle problem in terms of Eulerian cycles. Notes: Hamiltonian cycles - circuits that visit each vertex of a graph exactly once - apparently first arose in Euler's study of the knight's tour problem, although they were popularized by Hamilton's ``Around the World'' game in 1839. www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE176.HTM   (779 words)

 Citations: phase transition is not hard for the Hamiltonian Cycle problem - Vandegriend, Culberson (ResearchIndex)   (Site not responding. Last check: 2007-10-09) For the Hamiltonian cycle problem which is NP compete, Komlos and Szemeredi [14] not only proved the existence of the phase transition in this problem but also gave the exact location of the transition point. These problems are not so interesting as the NP complete problems from a complexity theoretic point of view because they can be solved in polynomial time. We establish two random models for the decision problem of NK landscapes and study the threshold phenomena and the associated hardness of the phase transitions in these two.... citeseer.ist.psu.edu /context/867122/125945   (866 words)

 [No title] Recall: A Hamiltonian cycle in a graph exists if there is a path that reaches every node and returns to the starting point as long as no node (except the first) is reached more than once and no edge is used more than once. Reduction#1: (problem 13.10) show that the Hamiltonian cycle problem for undirected graphs is reducible to the Hamiltonian cycle problem for directed graphs. Reduction #2: (problem 13.11) show that the Hamiltonian cycle problem is reducible to the traveling salesman problem. www.nku.edu /~foxr/CSC464/PROBLEMS/prob13-reductions.doc   (352 words)

 The Traveling Salesman Problem Problem: Given a complete undirected graph G=(V, E) that has nonnegative integer cost c(u, v) associated with each edge (u, v) in E, the problem is to find a hamiltonian cycle (tour) of G with minimum cost. Note that the TSP problem is NP-complete even if we require that the cost function satisfies the triangle inequality. Equivalently, if any problem in NP is not polynomial-time solvable than no NP-complete problem is polynomial solvable. www.personal.kent.edu /~rmuhamma/Algorithms/MyAlgorithms/AproxAlgor/TSP/tsp.htm   (639 words)

 Traveling Salesman Problem   (Site not responding. Last check: 2007-10-09) The problem is restricted to the Euclidian case where the TSP can be formulated as follows: Given n cities in the plane and their Euclidian distances, the problem is to find the shortest TSP-tour, i.e. Circulant Travelling Salesman Problem (CTSP) is the problem of finding a minimum weight Hamiltonian cycle in a weighted graph with circulant distance matrix. TSPLIB is a library of sample instances for the TSP (and related problems) from various sources and of various types. www.cis.njit.edu /~czumaj/TEACHING/CIS786/Fall2002/Homework1/TSP.html   (1989 words)

 1.5.5 Hamiltonian Cycle   (Site not responding. Last check: 2007-10-09) 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. Closely related is the problem of finding the longest path or cycle in a graph, which occasionally arises in pattern recognition problems. www.cs.sunysb.edu /~algorith/files/hamiltonian-cycle.shtml   (195 words)

 No Title Problem A is said to be polynomial-time reducible to Problem B if, for some fixed integer k, the following is true. As an example, we show that the Eulerian cycle problem is polynomial-time reducible to the Hamiltonian cycle problem. V=n be the input to the Eulerian cycle problem. www.cs.princeton.edu /courses/archive/fall97/cs341/handouts/16/16.html   (413 words)

 Hamiltonian Cycles   (Site not responding. Last check: 2007-10-09) The Hamiltonian cycle problem is one of the most famous in graph theory. If G does not have a Hamiltonian cycle, then there can be no such TSP tour in G', because the only way to get a tour of cost n in G would be to use only edges of weight 1, which implies a Hamiltonian cycle in G. Since the latter is the case, this reduction shows that TSP is hard, at least as hard as Hamiltonian cycle. www.cs.toronto.edu /~yuana/AlgorithmManual/BOOK/BOOK3/NODE107.HTM   (335 words)

 Charles Babbage Institute: RESEARCH PROGRAM> Current research   (Site not responding. Last check: 2007-10-09) The problem is named after Sir William Rowan Hamilton, an Irish mathematician who studied the problem in the 1850s. The problem was also reexamined by the Viennese mathematician Karl Menger in the 1920s, and Princeton’s Hassler Whitney and Merrill Flood in the 1930s. The Hamiltonian cycle problem is a variant of the traveling salesperson problem, which asks, “Given a map of N cities connected by roads, can a salesperson visit all the cities exactly once within some number of miles?” The traveling salesperson problem is also infeasible. www.cbi.umn.edu /shp/entries/hamiltoniancycleproblem.html   (255 words)

 ipedia.com: Hamiltonian path Article   (Site not responding. Last check: 2007-10-09) A Hamiltonian cycle is a cycle that visits each vertex exactly once, except for the starting vertex. 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)

 Polynomial Algorithms for Hamiltonian Cycle in Cocomparability Graphs   (Site not responding. Last check: 2007-10-09) In this paper the authors settle the complexity of the Hamiltonian problem in the more general class of cocomparability graphs. It is shown that the Hamiltonian cycle existence problem for cocomparability graphs is in \$P\$. The approach is based on exploiting the relationship between the Hamiltonian problem in a cocomparability graph and the bump number problem in a partial order corresponding to the transitive orientation of its complementary graph. epubs.siam.org /sam-bin/dbq/article/20037   (167 words)

 Artifact 627   (Site not responding. Last check: 2007-10-09) The red edges are forced to be in any Hamiltonian cycle because they are each incident on a vertex of degree two. If the blue edge is added to the current Hamiltonian cycle, then the fl edges incident to the blue vertex on the right must all be pruned. This makes the left blue vertex degree one, which implies the Hamiltonian cycle cannot be completed. www.cs.cmu.edu /afs/cs/project/jair/pub/volume9/vandegriend98a-appendix2/append.html   (277 words)

 TSPLIB   (Site not responding. Last check: 2007-10-09) This problem is an asymmetric traveling salesman problem with additional constraints. The problem is to find tours for the trucks of minimal total length that satisfy the node demands without violating truck capacity constraint. Except for the Hamiltonian cycle problems, all problems instances are defined on a complete graph and, at present, all distances are integer numbers. elib.zib.de /pub/Packages/mp-testdata/tsp/tsplib/tsplib.html   (376 words)

 Lecture 23--shortest path algorithms; transitive closure. 11/19/97. For each problem we have given one or more algorithms which solve the problem and for each algorithm we have give estimates of the time and space required for the algorithm to run in terms of the size of the input. The problems we have looked at so far can all be solved in time which is polynomial in their input size. We define the class NP to be the class of problems for which it is possible to decide in polynomial time whether or not a proposed solution is valid. www.ececs.uc.edu /~cpurdy/lec332_11.html   (1522 words)

 Charles Babbage Institute: RESEARCH PROGRAM> Current research   (Site not responding. Last check: 2007-10-09) As a consequence, the largest particular instance of one variant of the problem called the Hamiltonian cycle problem that has been solved to date connects 15,112 cities in Germany over approximately 66,000 kilometers. The earliest reference to the “traveling salesman problem” is not known, but is generally presumed a term coined Hassler Whitney of Princeton University during the 1931-32 academic year. At the heart of the traveling salesperson problem is what is sometimes called the “curse of dimensionality,” as a thoroughly exhaustive search for the optimal route through N cities would require checking (N-1)! www.cbi.umn.edu /shp/entries/travelingsalesperson.html   (323 words)

 2000mid2 The decision problem is to take a graph G and decide if G contains a Hamiltonian cycle. So you must show that if the decision problem has a polynomial time algorithm then the optimization problem also has a polynomial time algorithm. Recall that a Hamiltonian cycle is a cycle that visits every vertex exactly once. www.cs.pitt.edu /~kirk/cs1510/exams/2000mid2/2000mid2.html   (758 words)

 Project Suggestions   (Site not responding. Last check: 2007-10-09) Hamiltonian cycles and paths in Cayley Graphs: The "minimum change" graphs on permutations that we studied in class are all examples of Cayley Graphs. We are interested in finding algorithms and heuristics to solving the Hamiltonian cycle problem quickly in Cayley graphs. Graph Isomorphism: A fundamental open problem in graph theory is determining whether a given pair of graphs are isomorphic. www.cs.uiowa.edu /~sriram/196/fall01/projects.html   (304 words)

 The Hamiltonian Cycle Problem. A Hamiltonian cycle c of G is a cycle that goes through every vertex exactly once. (The problem with the covering graphs is addressed in the next section.) Gel electrophoresis is a technique that discriminates molecules of different sizes and different geometry and topology of DNA ([3,7,36,37]). By gel electrophoresis we select molecules that are mn in length, where m is the number of vertices in the graph and n is the length of the single strands used for the vertex building blocks. www.math.usf.edu /~jonoska/bio-comp/node3.html   (893 words)

 No Title   (Site not responding. Last check: 2007-10-09) You may use the fact that the Hamiltonian Cycle is NP-Complete to assist your proof: the Hamiltonian Cycle problem determines if there exists a simple cycle in a graph visiting each vertex exactly once. The King Arthur's problem is NP-hard: We reduce the Hamiltonian Cycle problem to the King Arthur's problem. Given an instance of the Hamiltonian Cycle problem, that is, a graph G, we take each vertex as a knight, and determine that two knights are enemies of each other iff they represent vertices that are not connected by an edge. ranger.uta.edu /~cook/aa/hw/qs13/qs13.html   (325 words)

 Finding Hamiltonian Cycles: Algorithms, Graphs and Performance   (Site not responding. Last check: 2007-10-09) The problem consists of finding a tour (or cycle) that visits all the vertices once and returns to the starting vertex. Generalizations of the knight's tour problem (a subset of the Hamiltonian cycle problem). My goal here is to identify graphs and graph properties that make the Hamiltonian cycle problem hard or that distinguish between various Hamiltonian cycle algorithms. web.cs.ualberta.ca /~joe/Theses/vandegriend.html   (90 words)

 A Hybrid Neural Network Model for Solving Optimization Problems   (Site not responding. Last check: 2007-10-09) An energy function which contains the constraints and cost criteria of an optimization problem is derived, and then the neural network is used to find the global minimum (or maximum) of the energy function, which corresponds to a solution of the optimization problem. The constraint network models the constraints of an optimization problem and computes the gradient (updating) value of each neuron such that the energy function monotonically converges to satisfy all constraints of the problem. The traveling salesman problem and the Hamiltonian cycle problem are used to demonstrate the method. csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/trans/tc/&toc=comp/trans/tc/1993/02/t2toc.xml&DOI=10.1109/12.204794   (779 words)

 Algorithms -- May '96   (Site not responding. Last check: 2007-10-09) Given a graph G with positive and negative edge weights and two designated vertices s and t, the negative s-t path problem is whether there exists some simple path between s to t whose weight is negative (the weight of a path is the sum of the weights of the edges on the path). Recall that the Hamiltonian cycle problem is whether, given an arbitrary graph, there exists a cycle which includes all the vertices. You have to show that the negative s-t path problem is NP-hard by finding a polynomial-time reduction from Hamiltonian cycle to it i.e. www.cs.rice.edu /CS/CSGSA/DeptInfo/Exams/alg-m96.html   (377 words)

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