12 research outputs found
A probabilistic polynomial algorithm for solving a directed Hamiltonian Path problem
No full textAbstract: "In this paper we present a graph-theoretic polynomial algorithm which has positive probability of finding a Hamiltonian Path in a given graph, if there is one; if the algorithm fails, it can be rerun with a randomly chosen starting solution, and there is again a positive probability it will find an answer. If there is no Hamiltonian Path, the algorithm will always terminate with failure. Some basic theoretical results concerning spanning arborescences of a graph are given. The concept of a ramification index is defined and it is shown that ramification index of a Hamiltonian Path is zero. The algorithm starts with finding any arborescence and by suitable pivots it endeavors to reduce the ramification index to zero. Probabilistic properties of the algorithm are discussed. Computational experience with graphs up to 30,000 nodes is included.
A successful algorithm for the undirected Hamiltonian path problem
No full textTepper School of Busines
A dynamic space-time network flow model for city traffic congestion
No full textAbstract: "A space-time network is used to model traffic flows over time for a capacitated road transportation system having one-way and two-way streets. Also, for the first time, traffic signal lights which change the network structure are explicitly incorporated into the model. A linear (time) cost per unit flow is associated with each arc, and it is shown that under the model structure, travel time on a street is a piecewise linear convex function of the number of units traveling on that street. Hence congestion effects are explicitly considered while maintaining the linear nature of the model.Two efficient solution methods are proposed. A network flow solution for a multiple source single destination network and a shortest path solution for a single source single destination network. Two examples are presented. The first example has one source and one sink. There is a unimodal buildup of traffic at the source (say a factory) which enters the street network as quickly as its capacity permits and proceeds through the network, stopping at red lights when necessary, towards the sink (a residential area). Computations with this example show that the arrival rate has multiple peaks which are induced by the stop lights. In the second example there are multiple sources and one sink. The results here are similar except that the arrival rate has a single broad peak which is due to the extreme symmetry of the constraints of the problems.
Solution of constrained generalized transportation problems using the pivot and probe algorithm
No full textAbstract: "In this paper we use a specialized version of our pivot and probe algorithm to solve generalized transportation problems with side constraints. The dual of an m x n generalized transportation problem with t side constraints is a linear program with m + n + t variables and up to m x n constraints. We solve the dual problem using the probe operation to select only the most important constraints to consider. We present computational experience on problems of sizes up to 180 x 180, having various degrees of density and having as many as 10 side constraints. It was found that for a given size and density, problems become harder to solve as the number of side constraints increases. Also, for a fixed number of side constraints, the solution difficulty increases with size and density. We found that our method was able to solve problems of the quoted sizes relatively quickly, with relatively few pivots, and without using basis reinversion.
