Showing papers on "Longest path problem published in 1982"

Hamilton Paths in Grid Graphs

Abstract: A grid graph is a node-induced finite subgraph of the infinite grid. It is rectangular if its set of nodes is the product of two intervals. Given a rectangular grid graph and two of its nodes, we give necessary and sufficient conditions for the graph to have a Hamilton path between these two nodes. In contrast, the Hamilton path (and circuit) problem for general grid graphs is shown to be NP-complete. This provides a new, relatively simple, proof of the result that the Euclidean traveling salesman problem is NP-complete.

Distributed computation on graphs: shortest path algorithms

Abstract: We use the paradigm of diffusing computation, introduced by Dijkstra and Scholten, to solve a class of graph problems. We present a detailed solution to the problem of computing shortest paths from a single vertex to all other vertices, in the presence of negative cycles.

Matrices eigenvalues, and directed graphs

Abstract: We show how several classical results concerning inclusion regions and estimates for the eigenvalues of matrices can be unified and generalized by the use of directed graphs. Applications to nonnegative matricesM-matrices, and the spectra of graphs are given.

An algorithm for the steiner problem in graphs

Abstract: The Steiner problem in graphs is concerned with finding a set of edges with minimum total weight which connects a given subset of points in a weighted graph. A branch and bound algorithm for solving this problem is presented together with an interesting application to a problem in molecular evolution. Computational experience gained in using the algorithm compares favorably, for certain classes of graphs, with that of existing methods.

The Bounded Path Tree Problem

Abstract: The subject of this paper is the bounded path tree (BPT) problem: An undirected graph $G ( V,E )$ is given whose edges have nonnegative lengths; two subsets I and J of V are also given, and nonnegative constants $U_i $, $W_i $ are associated with each $i \in I$, $j \in J$. The BPT problem asks for a tree of G whose vertex set contains $I \cup J$ and whose path joining vertices i and j is not longer than $U_i + W_j $, for each $i \in I$, $j \in J$. This problem generalizes the shortest path and the minimum longest path spanning tree problem. It complements standard min–max location problems, as it asks for a tree given the facility locations, instead of locating facilities in a given network. In this paper we propose some applications of the BPT problem for the design of emergency and communication networks, show its equivalence to an extension of the absolute center location problem and give an algorithm for its solution. This algorithm requires time $O( k| E | + k | V | \log k )$, where $k = | I \cup J |...

Some shortest path algorithms with decentralized information and communication requirements

Abstract: This paper presents several decentralized algorithms for finding all shortest paths in a network. Both static and dynamic networks are discussed. Each algorithm has localized information and communication requirements, operates asynchronously, and converges to the optimum in finite time.

Geographic Decomposition of the Shortest Path Problem, with an Application to the Traffic Assignment Problem

Abstract: An algorithm, which can be applied to loosely connected networks, is given for geographically decomposing the shortest path problem. The algorithm is applicable to the traffic assignment problem when it is solved as a series of shortest path problems by the Frank-Wolfe algorithm. Numerical results for a large 1287 node, 3752 arc traffic assignment problem for Washington, D.C., indicate that using geographical decomposition can reduce computer memory storage requirements or program run time.

Time minimizing flows in directed networks

Abstract: An important class of network flow problems is that class for which the objective is to minimize the cost of the most expensive unit of flow while obtaining a desired total flow through the network. Two special cases of this problem have been solved, namely, the bottleneck assignment problem and time-minimizing transportation problem. This paper addresses the more general case which we shall refer to as the time-minimizing network flow problem. Associated with each arc is an arc capacity (static) and a transferral time. The objective is to find a maximal flow for which the length (in time) of the longest path carrying flow is minimized. The character of the problem is discussed and a solution algorithm is presented.