The shortest path problem with two objective functions

We present methods to find the shortest path in a network where each path is associated with two objectives. We describe how to obtain the nondominated paths and the extreme nondominated paths, and compare the expected complexity of the methods. An improvement in efficiency can be obtained when quasiconcave or quasiconvex utility functions are assumed. In the first case, we describe how to find the optimal extreme nondominated path and bounds for the optimal path value. Then the optimal path can be located by calculating the k-th shortest path. In the second case we suggest a branch and bound method to solve the problem.     

An efficient algorithm for the Steiner Tree Problem with revenue,bottleneck and hop objective functions

This paper considers a tricriteria Steiner Tree Problem arising in the design of telecommunication networks. The objective functions consist of maximizing the revenue and of minimizing the maximal distance between each pair of interconnected nodes, as well as the maximal number of arcs between the root and each node. A polynomial algorithm is developed for the generation of a minimal complete set of Pareto-optimal Steiner trees. Optimality proofs are given and computational experience on a set of randomly generated problems is reported.     

Algorithms for time-dependent bicriteria shortest path problems

In this paper we generalize the classical shortest path problem in two ways. We consider two objective functions and time-dependent data. The resulting problem, called the time-dependent bicriteria shortest path problem (TdBiSP), has several interesting practical applications, but has not gained much attention in the literature. After reviewing relevant literature we develop a new algorithm for the TdBiSP with non-negative data. Numerical tests show the superiority of our algorithm compared with an existing algorithm in the literature. Furthermore, we discuss algorithms for the TdBiSP with negative travel times and costs.     

A transformation of hard (equality constrained) knapsack problems into constrained shortest path problems

The purpose of this paper is to show how hard (equality constrained) knapsack problems may be converted into constrained shortest path problems. Several directions along which such a transformation might be exploited for algorithmic purposes are suggested.     

Sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs

This paper addresses sensitivity analysis questions concerning the shortest path problem and the maximum capacity path problem in an undirected network. For both problems, we determine the maximum and minimum weights that each edge can have so that a given path remains optimal. For both problems, we show how to determine these maximum and minimum values for all edges in O(m + K log K) time, where m is the number of edges in the network, and K is the number of edges on the given optimal path.     

Speeding up the Floyd-Warshall algorithm for the cycled shortest path problem

On a network with a cycle, where at least one cycle exists, the Floyd-Warshall algorithm is one of the algorithms most used for determining the least cost path between every pair of nodes. In this work a new algorithm for this problem is developed that requires less computational effort than the Floyd-Warshall algorithm. Furthermore, we show that the basis of our algorithm is much easier to understand, which might be an advantage for educational purposes. A small example validates our algorithm and shows its implementation.     

A fuzzy shortest path with the highest reliability

This paper concentrates on a shortest path problem on a network where arc lengths (costs) are not deterministic numbers, but imprecise ones. Here, costs of the shortest path problem are fuzzy intervals with increasing membership functions, whereas the membership function of the total cost of the shortest path is a fuzzy interval with a decreasing linear membership function. By the max–min criterion suggested in [R.E. Bellman, L.A. Zade, Decision-making in a fuzzy environment, Management Science 17B (1970) 141–164], the fuzzy shortest path problem can be treated as a mixed integer nonlinear programming problem. We show that this problem can be simplified into a bi-level programming problem that is very solvable. Here, we propose an efficient algorithm, based on the parametric shortest path problem for solving the bi-level programming problem. An illustrative example is given to demonstrate our proposed algorithm.     

Multiobjective shortest path problems with lexicographic goal-based preferences

Multiobjective shortest path problems are computationally harder than single objective ones. In particular, execution time is an important limiting factor in exact multiobjective search algorithms. This paper explores the possibility of improving search performance in those cases where the interesting portion of the Pareto front can be initially bounded. We introduce a new exact label-setting algorithm that returns the subset of Pareto optimal paths that satisfy a set of lexicographic goals, or the subset that minimizes deviation from goals if these cannot be fully satisfied. Formal proofs on the correctness of the algorithm are provided. We also show that the algorithm always explores a subset of the labels explored by a full Pareto search. The algorithm is evaluated over a set of problems with three objectives, showing a performance improvement of up to several orders of magnitude as goals become more restrictive.     

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     

Martins' algorithm revisited for multi-objective shortest path problems with a MaxMin cost function

This paper presents a direct extension of the label setting algorithm proposed by Martins in 1984 for the shortest path problem with multiple objectives. This extended version computes all the efficient paths from a given source vertex, to all the other vertices of the network. The algorithm copes with problems in which the "cost values" associated with the network arcs are positive. The proposed extension can handle objective functions that are either of the "sum" type or of the "bottleneck" type. The main modifications to Martins' algorithm for multi-objective shortest path problems are linked to the dominance test and the procedure for identifying efficient paths. The algorithmic features are described and a didactic example is provided to illustrate the working principle. The results of numerical experiments concerning the number of efficient solutions produced and the CPU time consumed for several configurations of objectives, on a set of randomly generated networks, are also provided. Received: February 2005 / Revised version: June 2005 AMS classification: 90C29, 90C27, 05C38, 90B18, 68M12     

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.     

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.     

A column generation method for inverse shortest path problems

In this paper we formulate an inverse shortest path problem as a special linear programming problem. A column generation scheme is developed that is able to keep most columns of the LP model implicit and to generate necessary columns by shortest path algorithms. This method can get an optimal solution in finitely many steps. Some numerical results are reported to show that the algorithm has a good performance.The authors gratefully acknowledge the partial support of Croucher Foundation.     

最短路博弈群体单调分配方案构造

群体单调分配方案(Population Monotonic Allocation Scheme, 后简称PMAS)是合作博弈的一类分配机制。在合作博弈中, PMAS为每一个子博弈提供一个满足群体单调性的核中的分配方案, 从而保证大联盟的动态稳定性。本文主要贡献为利用线性规划与对偶理论构造与求解一类基于最短路问题的合作博弈(最短路博弈)的PMAS。我们首先借助对偶理论, 利用组合方法为最短路博弈构造了一个基于平均分摊思想的PMAS。然后借鉴计算核仁的Maschler方案, 将PMAS的存在性问题转化为一个指数规模的线性规划的求解问题, 并通过巧妙的求解得到了与之前组合方法相同的最短路博弈的PMAS。     

最短路博弈群体单调分配方案构造

群体单调分配方案(Population Monotonic Allocation Scheme, 后简称PMAS)是合作博弈的一类分配机制。在合作博弈中, PMAS为每一个子博弈提供一个满足群体单调性的核中的分配方案, 从而保证大联盟的动态稳定性。本文主要贡献为利用线性规划与对偶理论构造与求解一类基于最短路问题的合作博弈(最短路博弈)的PMAS。我们首先借助对偶理论, 利用组合方法为最短路博弈构造了一个基于平均分摊思想的PMAS。然后借鉴计算核仁的Maschler方案, 将PMAS的存在性问题转化为一个指数规模的线性规划的求解问题, 并通过巧妙的求解得到了与之前组合方法相同的最短路博弈的PMAS。     

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.     

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.     

Solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs

In this paper, we present an algorithm for solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs. The path cost function considered is comprised of two attributes, travel time and toll, that are combined into a nonlinear generalized cost. Travel demand is determined endogenously according to a travel disutility function. Travelers choose routes with the minimum overall generalized costs. The algorithm involves two components: a bicriteria shortest path routine to implicitly generate the set of non-dominated paths and a projection and contraction method to solve the nonlinear complementarity problem (NCP) describing the traffic equilibrium problem. Numerical experiments are conducted to demonstrate the feasibility of the algorithm to this class of traffic equilibrium problems.