: In this paper, the optimal path problem will be studied from a global point of view and having no restrictions imposed on the network. The concepts of boundness and finiteness will be presented for the general problem and will be studied in two particular cases. Special emphasis will be given to the optimality principle since it allows one to design a class of algorithms -- the labelling algorithms -- which determine an optimal path when the weak optimality principle is satisfied. Its importance will be stressed by two problems which, in its turn, are similar in their description and completly different in what concerns their resolution. Keywords: network, path, optimality principle, labelling algorithms. 1 Introduction In the optimal path problem, a real function is considered which assigns a value to each path that can be defined between a given pair of nodes in a given network; a path with the best value in a subset of paths between that pair of nodes is what has to be determin...

The shortest path problem is a classical network problem that has been extensively studied. The problem of determining not only the shortest path, but also listing the K shortest paths (for a given integer K ? 1) is also a classical one but has not been studied so intensively, despite its obvious practical interest. Two different types of problems are usually considered: the unconstrained and the constrained K shortest paths problem. While in the former no restriction is considered in the definition of a path, in the constrained K shortest paths problem all the paths have to satisfy some condition -- for example, to be loopless. In this paper new algorithms are proposed for the unconstrained problem, which compute a super set of the K shortest paths. It is also shown that ranking loopless paths does not hold in general the Optimality Principle and how the proposed algorithms for the unconstrained problem can be adapted for ranking loopless paths. Keywords: Network, tree, path, path d...

: In this paper the ranking of shortest paths problem is viewed as a generalization of the shortest path problem, since the K shortest paths have to be computed. Finitness and boundness are studied as well as conditions are established in order that an Optimality Principle generalization is satisfied. Under these conditions the K shortest paths can be determined by a natural generalization of forms of the labeling algorithm for the shortest path problem. This generalization is the main subject of the paper. Keywords: network, path, path cost, paths ranking, optimality principle, labeling. 1 Introduction In this paper the ranking of shortest paths is considered as a generalization of the well known shortest path problem, [5, 6, 7, 8, 9, 10, 12], since several paths must be determined. In fact, in the ranking of shortest paths problem, for a given integer K (K 1), the K shortest paths between a given pair of nodes have to be listed by nondecreasing order of their costs. Sometimes K de...

In this note we are concerned with the ranking of shortest loopless paths problem. It is shown that in general this problem does not satisfy the optimality principle and as a consequence only methods based on the computation of a super set of the set of the K shortest loopless paths are valid. It is shown how this methods for the unconstrained problem can be adapted for determining the K shortest loopless paths.