Ernesto de Queirós Vieira Martins

On a multicriteria shortest path problem

Abstract Multicriteria shortest path problems have not been treated intensively in the specialized literature, despite their potential applications. In fact, a single objective function may not be sufficient to characterize a practical problem completely. For instance, in a road network several parameters (as time, cost, distance, etc.) can be assigned to each arc. Clearly, the shortest path may be too expensive to be used. Nevertheless the decision-maker must be able to choose some solution, possibly not the best for all the criteria. In this paper we present two algorithms for this problem. One of them is an immediate generalization of the multiple labelling scheme algorithm of Hansen for the bicriteria case. Based on this algorithm, it is proved that any pair of nondominated paths can be connected by nondominated paths. This result is the support of an algorithm that can be viewed as a variant of the simplex method used in continuous linear multiobjective programming. A small example is presented for both algorithms.

A new implementation of Yen’s ranking loopless paths algorithm

Abstract.Yen’s algorithm is a classical algorithm for ranking the K shortest loopless paths between a pair of nodes in a network. In this paper an implementation of Yen’s algorithm is presented. Both the original algorithm and this implementation present

A bicriterion shortest path algorithm

{\cal O}(Kn(m + n\log n))

An algorithm for the ranking of shortest paths

computational complexity order when considering a worst-case analysis. However, computational experiments are reported, which allow to conclude that in practice this new implementation outperforms two other, Perko’s implementation and a straightforward one.

DEVIATION ALGORITHMS FOR RANKING SHORTEST PATHS

Abstract Among the network models, one of the more popular is the so called shortest path problem. This model is used whenever it is intended to minimize a linear function which represents a distance between a predetermined pair of nodes in a given network. Often a single objective function is not sufficient to completely characterize some real-world problems. For instance, in a road network two parameters - as cost and time - can be assigned to each arc. Clearly the fastest path may be too costly. Nevertheless the decision-maker must choose one solution, possibly not the best for both criteria. In this paper we present an algorithm for this problem. With this algorithm a special set of paths (the set of Pareto optimal paths) is determined. One objective for any Pareto optimal path can not be improved without worsening the other one.

An algorithm for the quickest path problem

An efficient computational implementation of a path deletion K shortest paths algorithm and a new algorithm for the same problem are presented. In a path deletion K shortest paths algorithm a sequence {g1, g2,…, gK} of networks is is defined, such that g1 is given network and its k-th shortes path is trivially determined from the shortest path in gk. In essence, as soon as the shortest path in gK is determined it is excluded from gk in such a way that no new paths are formed and no more paths are deleted. So, far each gk two procedures are executed: a shortest path algorithm and a path deletion algorithm. In the presented computational implementation, all the information resulting from the determination of the k-th shortest path is carried throughout gk + 1, gk + 2,…, gK. The new algorithm not only keeps this characteristics but also avoids the last K−1 executions of a shortest path algorithm, which results in a suprising and very substantial reduction in the execution time. In fact, for randomly generated networks with 104 nodes and 105 arcs, once the shortest the new algorithm computes the next 100 shortest paths in times of the order of 10−1 seconds. To illustrate the efficiency of this algorithm, comparative computational experiments are reported.

An algorithm for ranking paths that may contain cycles

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 in 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 uncontrained 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.

Solving bicriteria 0-1 knapsack problems using a labeling algorithm

The quickest path problem arises when transmitting a given amount of data between two given nodes of a network, a lead time and a capacity (per unit of time) being assigned to each arc in the network. In this paper the problem is regarded as a bicriteria path problem, allowing the use of a very efficient algorithm which solves the quickest path problem for all possible values of the amount of data that has to be transmitted.

A computational improvement for a shortest paths ranking algorithm

Abstract In this paper an algorithm is presented for determining the K best paths that may contain cycles in a directed network. The basic idea behind the algorithm is quite simple. Once the best path has been determined it is excluded from the network in such a way that no new path is formed and no more paths are excluded. This step leads to an enlarged network where all the paths, but the best one, can be determined. The method is repeated until the desired paths have been computed. The proposed algorithm can be used not only for the classical K shortest paths problem but also for ranking paths under a nonlinear objective function, provided that an algorithm to determine the best path exists. Computational results are presented and comparisons with other approaches for the classical problem are made.

On a special class of bicriterion path problems

This paper examines the performances of a new labeling algorithm to find all the efficient paths (or non-dominated evaluation vectors) of the bicriteria 0-1 knapsack problem. To our knowledge this is the first time a bicriteria 0-1 knapsack is solved taking advantage of its previous conversion into a bicriteria shortest path problem over an acyclic network. Computational experiments and results are also presented regarding bicriteria instances of up to 900 items. The algorithm is very efficient for the hard bicriteria 0-1 knapsack instances considered in the paper.