The tricriterion shortest path problem with at least two bottleneck objective functions

The focus of this paper is on the tricriterion shortest path problem where two objective functions are of the bottleneck type, for example MinMax or MaxMin. The third objective function may be of the same kind or we may consider, for example, MinSum or MaxProd. Let p(n) be the complexity of a classical single objective algorithm responsible for this third function, where n is the number of nodes and m be the number of arcs of the graph. An O(m²p(n)) algorithm is presented that can generate the minimal complete set of Pareto-optimal solutions. Finding the maximal complete set is also possible. Optimality proofs are given and extensions for several special cases are presented. Computational experience for a set of randomly generated problems is reported.     

The shortest path problem with forbidden paths

We consider a variant of the constrained shortest path problem, where the constraints come from a set of forbidden paths (arc sequences) that cannot be part of any feasible solution. Two solution approaches are proposed for this variant. The first uses Aho and Corasick's keyword matching algorithm to filter paths produced by a k-shortest paths algorithm. The second generalizes Martins' deviation path approach for the k-shortest paths problem by merging the original graph with a state graph derived from Aho and Corasick's algorithm. Like Martins' approach, the second method amounts to a polynomial reduction of the shortest path problem with forbidden paths to a classic shortest path problem. Its significant advantage over the first approach is that it allows considering forbidden paths in more general shortest path problems such as the shortest path problem with resource constraints.     

The dynamic shortest path problem with anticipation

Mobile communication technologies enable truck drivers to keep abreast of changing traffic conditions in real-time. We assume that such communication capability exists for a single vehicle traveling from a known origin to a known destination where certain arcs en route are congested, perhaps as the result of an accident. Further, we know the likelihood, as a function of congestion duration, that congested arcs will become uncongested and thus less costly to traverse. Using a Markov decision process, we then model and analyze the problem of constructing a minimum expected total cost route from an origin to a destination that anticipates and then responds to changes in congestion, if they occur, while the vehicle is en route. We provide structural results and illustrate the behavior of an optimal policy with several numerical examples and demonstrate the superiority of an optimal anticipatory policy, relative to a route design approach that reflects the reactive nature of current routing procedures.     

Thek-centrum shortest path problem

Thek-Centrum Shortest Path Problem (kCSP[s, t]) is to minimize the sum of thek longest arcs in any (simple)s−t path containing at leastk arcs, wherek is a positive integer.kCSP is introduced and is shown to be NP-Hard, although it is polynomially solvable ifk is constrained to be no greater than the number of arcs in ans−t path with fewest arcs. Some properties of the problem are studied and a new weakly dual problem is also introduced.     

On a multicriteria shortest path problem

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.     

Solving the shortest path tour problem

In this paper, we study the shortest path tour problem in which a shortest path from a given origin node to a given destination node must be found in a directed graph with non-negative arc lengths. Such path needs to cross a sequence of node subsets that are given in a fixed order. The subsets are disjoint and may be different-sized. A polynomial-time reduction of the problem to a classical shortest path problem over a modified digraph is described and two solution methods based on the above reduction and dynamic programming, respectively, are proposed and compared with the state-of-the-art solving procedure. The proposed methods are tested on existing datasets for this problem and on a large class of new benchmark instances. The computational experience shows that both the proposed methods exhibit a consistent improved performance in terms of computational time with respect to the existing solution method.     

Distributionally robust stochastic shortest path problem

This paper considers a stochastic version of the shortest path problem, the Distributionally Robust Stochastic Shortest Path Problem(DRSSPP) on directed graphs. In this model, the arc costs are deterministic, while each arc has a random delay. The mean vector and the second-moment matrix of the uncertain data are assumed known, but the exact information of the distribution is unknown. A penalty occurs when the given delay constraint is not satisfied. The objective is to minimize the sum of the path cost and the expected path delay penalty. As it is NP-hard, we approximate the DRSSPP with a semidefinite programming (SDP for short) problem, which is solvable in polynomial time and provides tight lower bounds.     

The robust shortest path problem in series-parallel multidigraphs with interval data

In this paper the robust shortest path problem in edge series-parallel multidigraphs with interval costs is examined. The maximal regret criterion is applied to calculate the optimal solution. It is shown that this problem is NP-hard. A pseudopolynomial algorithm for the studied problem is constructed.     

The robust shortest path problem with interval data via Benders decomposition

Many real problems can be modelled as robust shortest path problems on digraphs with interval costs, where intervals represent uncertainty about real costs and a robust path is not too far from the shortest path for each possible configuration of the arc costs. In this paper we discuss the application of a Benders decomposition approach to this problem. Computational results confirm the efficiency of the new algorithm. It is able to clearly outperform state-of-the-art algorithms on many classes of networks. For the remaining classes we identify the most promising algorithm among the others, depending of the characteristics of the networks. Received: September 2004 / Accepted: March 2005 AMS classification: 90C47, 52B05, 90C57 The work was partially supported by the European Commission IST projects MOSCA (IST-2000-29557) and BISON (IST-2001-38923). All correspondence to: Roberto Montemanni     

A generalized shortest path tour problem with time windows

Computational Optimization and Applications - This paper studies a generalization of the shortest path tour problem with time windows (GSPTPTW). The aim is to find a single-origin...     

The undirected two disjoint shortest paths problem

The $k$ disjoint shortest paths problem ($k$-DSPP) on a graph with $k$ source–sink pairs $(s_i,t_i)$ asks if there exist $k$ pairwise edge- or vertex-disjoint shortest $s_i$–$t_i$-paths. It is known to be NP-complete if $k$ is part of the input. Restricting to $2$-DSPP with strictly positive lengths, it becomes solvable in polynomial time. We extend this result by allowing zero edge lengths and give a polynomial-time algorithm based on dynamic programming for $2$-DSPP on undirected graphs with non-negative edge lengths.     

New models for shortest path problem with fuzzy arc lengths

This paper considers the shortest path problem with fuzzy arc lengths. According to different decision criteria, the concepts of expected shortest path, α-shortest path and the most shortest path in fuzzy environment are originally proposed, and three types of models are formulated. In order to solve these models, a hybrid intelligent algorithm integrating simulation and genetic algorithm is provided and some numerous examples are given to illustrate its effectiveness.     

LP formulations of the shortest path tree problem

Based on a pair of primal-dual LP formulations of the shortest path tree problem, the first algorithmic approach to reoptimizing the shortest paths subject to changes in the edge weights was proposed by S. Pallottino and M.G. Scutellá in 2003. We shall here focus solely on their introductory sections, propose some simplifications of the models considered, and finally relate the resulting models to the presentation of single-source shortest path problems in textbooks treating this subject with but limited or no reference to LP.Received: April 2004, Revised: August 2004, MSC classification: 90C05, 90C35, 90B10 Dedicated to the memory of Stefano Pallottino     

Vector maximization with two objective functions

This note gives a characterization of an efficient solution for the vector maximization problem with two objective functions. This characterization yields a parametric procedure for generating the set of all efficient solutions for this problem. The parametric procedure can also be used to solve certain bicriterion mathematical programs.     

On the second point-to-point undirected shortest simple path problem

We address the determination of the second point-to-point shortest simple path in undirected networks. The effective reduced cost concept is introduced to compute the second best solution. This concept is used to prove that a path tree containing the second point-to-point shortest simple path is adjacent to any shortest path tree. Therefore, this result immediately implies a method requiring O(m) time once that the shortest path tree is obtained on an undirected network with n nodes and m edges.     

On the K shortest path trees problem

We address the problem of finding the K best path trees connecting a source node with any other non-source node in a directed network with arbitrary lengths. The main result in this paper is the proof that the kth shortest path tree is adjacent to at least one of the previous (k-1) shortest path trees. Consequently, we design an O(f(n,m,C_max)+Km) time and O(K+m) space algorithm to determine the K shortest path trees, in a directed network with n nodes, m arcs and maximum absolute length C_max, where O(f(n,m,C_max)) is the best time needed to solve the shortest simple paths connecting a source node with any other non-source node.