Topic
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.
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TL;DR: A graph theoretic version of Steiner's problem in plane geometry is described and it is shown that a solution to this problem provides us with a solutions to the problems of finding a minimum externally stable set and a maximum internally stable set in a graph.
Abstract: A graph theoretic version of Steiner's problem in plane geometry is described. An approach for solving this problem, related to Melzak's solution to Steiner's problem, is presented. The problems of finding “shortest route” and “minimal spanning tree” in graphs become special cases of the Steiner's problem in graphs. It is shown that a solution to this problem also provides us with a solution to the problems of finding a minimum externally stable set and a maximum internally stable set in a graph.
368 citations
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TL;DR: Eight algorithms which solve the shortest path tree problem on directed graphs are presented, together with the results of wide-ranging experimentation designed to compare their relative performances on different graph topologies.
Abstract: Theshortest path problem is considered from a computational point of view. Eight algorithms which solve theshortest path tree problem on directed graphs are presented, together with the results of wide-ranging experimentation designed to compare their relative performances on different graph topologies. The focus of this paper is on the implementation of the different data structures used in the algorithms. A "Pidgin Pascal" description of the algorithms is given, containing enough details to allow for almost direct implementation in any programming language. In addition, Fortran codes of the algorithms and of the graph generators used in the experimentation are provided on the diskette.
364 citations
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TL;DR: A new 0-1 linear programming formulation of the Project Scheduling Problem with resource constraints, corresponding to all feasible subsets of activities that can be simultaneously executed without violating resource or precedence constraints is presented.
Abstract: In this paper we consider the Project Scheduling Problem with resource constraints, where the objective is to minimize the project makespan. We present a new 0-1 linear programming formulation of the problem that requires an exponential number of variables, corresponding to all feasible subsets of activities that can be simultaneously executed without violating resource or precedence constraints. Different relaxations of the above formulation are used to derive new lower bounds, which dominate the value of the longest path on the precedence graph and are tighter than the bound proposed by Stinson et al. (1978). A tree search algorithm, based on the above formulation, that uses new lower bounds and dominance criteria is also presented. Computational results indicate that the exact algorithm can solve hard instances that cannot be solved by the best algorithms reported in the literature.
359 citations
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TL;DR: A favorable special case of the 3-D shortest path problem, namely that of finding the shortest path between two points along the surface of a convex polyhedron, is considered, which can be solved in time $O(n^3 \log n)$.
Abstract: We consider the problem of computing the shortest path between two points in two- or three-dimensional space bounded by polyhedral surfaces. In the 2-D case the problem is easily solved in time $O(n^2 \log n)$. In the general 3-D case the problem is quite hard to solve, and is not even discrete; we present a doubly-exponential procedure for solving the discrete subproblem of determining the sequence of boundary edges through which the shortest path passes. Finally we consider a favorable special case of the 3-D shortest path problem, namely that of finding the shortest path between two points along the surface of a convex polyhedron, and solve it in time $O(n^3 \log n)$.
323 citations
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TL;DR: This paper illustrates how the basic dynamic programming algorithm can be improved by bounded bi-directional search and experimentally evaluates the effectiveness of the enhancement proposed.
317 citations