Shortest path problem

In graph theory, the conversion of the adjacency matrix into the distance matrix is known as the "all pairs shortest path problem",... 

Dress, A.W.M. and Havel, T.F. Shortest path problems and molecular conformation. Discrete Appl. Math. 1988, 39, 129-144. 

V. Kumar and V. Singh,/. Parallel and Distributed Processing, 13, 124 (1991). Scalability of Parallel Algorithms for the All-Pairs Shortest Path Problem. 

A. W. M. Dress and T. F. Havel, Discrete Appl. Math., 19,129 (1988). Shortest-Path Problems... 

The Lagrangian dual problem (23)-(30) can be solved relatively easily because it decomposes into A constrained shortest-path problems, one for each OD pair. A constrained shortest-path problem... 

Steps 2-4 implement the Lagrangian releixation eilgorithm. In step 2, a constrained shortest-path problem is solved for each OD pair using the costs c = For the first iteration, the original... 

The linear programming formulation of the longest path problem is identical to that of the shortest path problem except that the objective function must be maximized instead of minimized. 

In order to solve the shortest path problem for the double weighting network, a new weight for graph G is constructed with the target weight and limit weight and which is expressed as w = aw, -H (1 - a) w and Wj = (k,W2 + fcj (0 < a < 1). If there is a feasible path p from the specified source node s to the destination node t, a function can be established as follows ... 

Current, J. et al. 1990. An interactive approach to inden-tify the best compromise solution for two objective Shortest path problems Computer Ops. Res., 17 (2) 187-192. 

R. Dorairaj and G. Lakhani. A VLSI implementation of all-pair shortest path problem. In ICPP 81 pages 207-209. S. K. Sahni, editor. The Pennsylvania State University Press, 1987. 

S. Y. Kung, P. S. Lewis, and S. C. Lo. Optimal systolic design for the transitive closure and shortest path problems. IEEE Trans, on Computers, C-36, number 5, pages 603-614, 1987. 

Figure 7.1. Shortest path problem to Example 7.1 shortest path tree is dashed.

The resulting shortest path problem is depicted in Figure 7.1. Reading off the shortest path distances to 1, 2 yiMs P = 1/4 and P2 = 5/A. So the auctioneer can realize revenue ofZ/A with this allocation rule. 

The second problem is the shortest path problem. Designate a source, s and sink t and let L A) be the length of the shortest s — t path using edges only in A Also to make things interesting, assume that the minimum 5 — t cut is at least two. 

There are many network systems in the world, for example, Internet, electricity network and traffic network, etc. These systems forms basis of society, so stabilizing operation of systems and designing systems to operate effectively are important problems. The evaluation of network systems considers various measures, for example, reliability, distances, flow and time. To optimize these measures, the problems are formulated as shortest path problem, maximum flow problem and so on. 

Optimization. In optimization, it is required to find a solution that optimizes an objective function subject to a set of constraints. Hopfield networks proved to be very effective in solving nonlinear optimization problems. Some examples are the application in the shortest path problem (Bousono-Carzon et al, 1997) and in combinatorial optimization problems (Colomi, etal, 1996). 

Buscono-Calzon, C. and Figueiras-Vidal, A.R. (1997) A bank of Hopfield neural networks for the shortest path problem. Journal of Computational and Applied Mathematics, 82 (1-2), 117-128. 

The algorithm that is going to be used for the final allocation of the students to the bus stops is going to be a variation of Dijkstra s algorithm that is used to solve the shortest path problem. In oiu ease we will use Dijkstra s algorithm in order to solve the safest path problem. 

As should be clear from the previous discussion, most papers on the routing of school buses have focused on capturing as many aspects of reahty as possible, building intricate multi-objective models with many complex constraints. On the other hand, we focus on a basic version of this problem to develop efficient methods that take the safety issue of the problem into accoimt. Our problem formulation has only a single school, one type of student and one type of bus, with fixed capacity and has been based upon the multi-trip elementary shortest path problem with resource constraints developed by Akca etal. [AKC 10]. 

AKC 10] Akca Z., Ralphs T.K., Berger R.T., Solution methods for the multi-trip elementary shortest path problem with resource constraints . Optimization Online the Mathematical Programming Society, 2010... 

R.K. Ahuja, K. Mehlhom, J.B. Orlin, R.E. Tarjan, Faster algorithms for the shortest path problem, J. Asso. Comput. Mach., 1990. 

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