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Longest path problem

About: Longest path problem is a research topic. Over the lifetime, 3264 publications have been published within this topic receiving 102814 citations.


Papers
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Journal ArticleDOI
01 Oct 1959
TL;DR: In this paper, it was shown that the length of the shortest closed path through n points in a bounded plane region of area v is almost always asymptotically proportional to √(nv) for large n; and this result was extended to bounded Lebesgue sets in k-dimensional Euclidean space.
Abstract: We prove that the length of the shortest closed path through n points in a bounded plane region of area v is ‘almost always’ asymptotically proportional to √(nv) for large n; and we extend this result to bounded Lebesgue sets in k–dimensional Euclidean space. The constants of proportionality depend only upon the dimensionality of the space, and are independent of the shape of the region. We give numerical bounds for these constants for various values of k; and we estimate the constant in the particular case k = 2. The results are relevant to the travelling-salesman problem, Steiner's street network problem, and the Loberman—Weinberger wiring problem. They have possible generalizations in the direction of Plateau's problem and Douglas' problem.

902 citations

Journal ArticleDOI
TL;DR: A simpleO(ND) time and space algorithm is developed whereN is the sum of the lengths of A andB andD is the size of the minimum edit script forA andB, and the algorithm performs well when differences are small and is consequently fast in typical applications.
Abstract: The problems of finding a longest common subsequence of two sequencesA andB and a shortest edit script for transformingA intoB have long been known to be dual problems. In this paper, they are shown to be equivalent to finding a shortest/longest path in an edit graph. Using this perspective, a simpleO(ND) time and space algorithm is developed whereN is the sum of the lengths ofA andB andD is the size of the minimum edit script forA andB. The algorithm performs well when differences are small (sequences are similar) and is consequently fast in typical applications. The algorithm is shown to haveO(N+D 2) expected-time performance under a basic stochastic model. A refinement of the algorithm requires onlyO(N) space, and the use of suffix trees leads to anO(N logN+D 2) time variation.

805 citations

Journal ArticleDOI
TL;DR: The set of pattern graphs for which the directed subgraph homeomorphism problem is NP-complete is characterized and a polynomial time algorithm is given for the remaining cases.

792 citations

Journal ArticleDOI
TL;DR: An approximate solution to the weighted-graph-matching problem is discussed for both undirected and directed graphs and an analytic approach is used instead of a combinatorial or iterative approach to the optimum matching problem.
Abstract: An approximate solution to the weighted-graph-matching problem is discussed for both undirected and directed graphs. The weighted-graph-matching problem is that of finding the optimum matching between two weighted graphs, which are graphs with weights at each arc. The proposed method uses an analytic instead of a combinatorial or iterative approach to the optimum matching problem. Using the eigendecompositions of the adjacency matrices (in the case of the undirected-graph-matching problem) or Hermitian matrices derived from the adjacency matrices (in the case of the directed-graph-matching problem), a matching close to the optimum can be found efficiently when the graphs are sufficiently close to each other. Simulation results are given to evaluate the performance of the proposed method. >

777 citations

Proceedings ArticleDOI
30 Apr 1974
TL;DR: This paper shows that a number of NP-complete problems remain NP- complete even when their domains are substantially restricted, and determines essentially the lowest possible upper bounds on node degree for which the problems remainNP-complete.
Abstract: It is widely believed that showing a problem to be NP-complete is tantamount to proving its computational intractability. In this paper we show that a number of NP-complete problems remain NP-complete even when their domains are substantially restricted. First we show the completeness of SIMPLE MAX CUT (MAX CUT with edge weights restricted to value 1), and, as a corollary, the completeness of the OPTIMAL LINEAR ARRANGEMENT problem. We then show that even if the domains of the NODE COVER and DIRECTED HAMILTONIAN PATH problems are restricted to planar graphs, the two problems remain NP-complete, and that these and other graph problems remain NP-complete even when their domains are restricted to graphs with low node degrees. For GRAPH 3-COLORABILITY, NODE COVER, and UNDIRECTED HAMILTONIAN CIRCUIT, we determine essentially the lowest possible upper bounds on node degree for which the problems remain NP-complete.

648 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202313
202240
202125
202028
201930
201844