A branch and bound algorithm for the robust shortest path problem with interval data

Many real problems can be modelled as robust shortest path problems on interval digraphs, 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.A branch and bound algorithm for this problem is presented.     

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     

Proportion-based robust optimization and team orienteering problem with interval data

In this paper, a proportion-based robust optimization approach is developed to deal with uncertain combinatorial optimization problems. This approach assumes that a certain proportion of uncertain coefficients in each solution are allowed to change and optimizes a deterministic model so as to achieve a trade-off between optimality and feasibility when the coefficients change. We apply this approach on team orienteering problem with interval data (TOPID), a variant of vehicle routing problem, which has not yet been studied before. A branch and price algorithm is proposed to solve the robust counterpart by using two novel dominance relations. Finally, numerical study is performed. The results show the usefulness of the proposed robust optimization approach and the effectiveness of our algorithm.     

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.     

Labeling algorithm for the shortest path problem with turn prohibitions with application to large-scale road networks

In real road networks, the presence of no-left, no-right or no U-turn signs, restricts the movement of vehicles at intersections. These turn prohibitions must be considered when calculating the shortest path between a starting and an ending point in a road network. We propose an extension of Dijkstra’s algorithm to solve the shortest path problem with turn prohibitions. The method uses arc labeling and a network structure with low memory requirements. We compare the proposed method with the dual graph approach in a set of randomly generated networks and Bogotá’s large-scale road network. Our computational experiments show that the performance of the proposed method is better than that of the dual graph approach, both in terms of computing time and memory requirements. We co-developed a Web-based decision support system for computing shortest paths with turn prohibitions that uses the proposed method as the core engine.     

New efficient shortest path simplex algorithm: pseudo permanent labels instead of permanent labels

We introduce a new network simplex pivot rule for the shortest path simplex algorithm. This new pivot rule chooses a subset of non-basic arcs to simultaneously enter into the basis. We call this operation a multiple pivot. We show that a shortest path simplex algorithm with this pivot rule performs O(n) multiple pivots and runs in O(nm) time. Our pivot rule is based on the new concept of a pseudo permanently labeled node, and it can be adapted to design a new label-correcting algorithm that runs in O(nm). Moreover, this concept lets us introduce new rules to identify negative cycles. Finally, we compare the network simplex algorithm with multiple pivots with other previously proposed efficient network simplex algorithm in a computational experiment.     

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.     

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.     

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.     

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.     

The shortest multipaths problem in a capacitated dense channel

In this paper, we present a simple polynomial-time algorithm solving the shortest multipaths problem in particular grid graphs called dense channels. Our work extends the results of Formann et al. [M. Formann, D. Wagner, F. Wagner, Routing through a dense channel with minimum total wire length, Journal of Algorithms 15 (1993) 267–283], by considering arbitrary horizontal and vertical capacities.     

On the complexity of the robust spanning tree problem with interval data

This paper studies the complexity of the robust spanning tree problem with interval data (RSTID). It shows that the problem is NP-complete, settling the conjecture of Kouvelis and Yu, and that it remains so for complete graphs or when the intervals are all [0,1]. These results indicate that the difficulty of RSTID stems from both the graph topology and the structure of the cost intervals, suggesting new directions for search algorithms.     

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.     

An aggregate label setting policy for the multi-objective shortest path problem

We consider label setting algorithms for the multi-objective shortest path problem with any number of sum and bottleneck objectives. We propose a weighted sum aggregate ordering of the labels, specifically tailored to combine sum and bottleneck objectives. We show that the aggregate order leads to a consistent reduction of solution times (up to two-thirds) with respect to the classical lexicographic order.     

Time version of the shortest path problem in a stochastic-flow network

Many studies on hardware framework and routing policy are devoted to reducing the transmission time for a flow network. A time version of the shortest path problem thus arises to find a quickest path, which sends a given amount of data from the unique source to the unique sink with minimum transmission time. More specifically, the capacity of each arc in the flow network is assumed to be deterministic. However, in many real-life networks, such as computer systems, telecommunication systems, etc., the capacity of each arc should be stochastic due to failure, maintenance, etc. Such a network is named a stochastic-flow network. Hence, the minimum transmission time is not a fixed number. We extend the quickest path problem to evaluating the probability that d$d$ units of data can be sent under the time constraint T$T$. Such a probability is named the system reliability. In particular, the data are transmitted through two minimal paths simultaneously in order to reduce the transmission time. A simple algorithm is proposed to generate all (d,T)$(d,T)$-MPs and the system reliability can then be computed in terms of (d,T)$(d,T)$-MPs. Moreover, the optimal pair of minimal paths with highest system reliability could be obtained.     

Two disjoint shortest paths problem with non-negative edge length

In the two disjoint shortest paths problem ( 2-DSPP), the input is a graph (or a digraph) and its vertex pairs $(s_1,t_1)$ and $(s_2,t_2)$, and the objective is to find two vertex-disjoint paths $P_1$ and $P_2$ such that $P_i$ is a shortest path from $s_i$ to $t_i$ for $i=1,2$, if they exist. In this paper, we give a first polynomial-time algorithm for the undirected version of the 2-DSPP with an arbitrary non-negative edge length function.     

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.     

Some tractable instances of interval data minmax regret problems

In this paper, we provide polynomial and pseudopolynomial algorithms for classes of particular instances of interval data minmax regret graph problems. These classes are defined using a parameter that measures the distance from well-known solvable instances. Tractable cases occur when the parameter is bounded by a constant.