A utility measure for finding multiobjective shortest paths in urban multimodal transportation networks

In this paper we study the problem of finding Origin–Destination (O–D) shortest paths in urban multimodal transportation networks, aiming at minimizing the overall cost, time and users' discommodity associated with the required paths. We present an approach based on the classical shortest path problem on a network representing the urban multimodal transportation system, i.e. the private, the public and the pedestrian modalities. Our idea is to make use of an ad hoc utility function for weighing the arcs both with their cost and time and considering at the same time the preference of the users related to all the possible transportation modalities. In particular, a utility measure is presented taking into a proper account the different users' propensities. The proposed approach has been developed for analysing the urban transportation network of an Italian city; the first experimental results are given in the paper.     

Dynamic programming algorithms for the elementary shortest path problem with resource constraints

When vehicle routing problems with additional constraints (e.g. capacities or time windows) are solved via column generation and branch-and-price, it is common that the pricing problem requires the computation of a minimum cost constrained path on a graph with costs on the arcs and prizes on the nodes. The pricing problem is usually solved via dynamic programming in two possible ways: requiring elementary paths or allowing paths with cycles. We experimentally compare these two strategies and we evaluate the effectiveness of some algorithmic ideas to improve their performance.     

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 quickest path problem with batch constraints

The quickest path problem has been proposed to cope with flow problems through networks whose arcs are characterized by both travel times and flowrate constraints. Basically, it consists in finding a path in a network to transmit a given amount of items from a source node to a sink in as little time as possible, when the transmission time depends on both the traversal times of the arcs and the rates of flow along arcs. This paper is focused on the solution procedure when the items transmission must be partitioned into batches with size limits. For this problem we determine how many batches must be made and what the sizes should be.     

Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints

When vehicle routing problems with additional constraints, such as capacity or time windows, are solved via column generation and branch-and-price, it is common that the pricing subproblem requires the computation of a minimum cost constrained path on a graph with costs on the arcs and prizes on the vertices. A common solution technique for this problem is dynamic programming. In this paper we illustrate how the basic dynamic programming algorithm can be improved by bounded bi-directional search and we experimentally evaluate the effectiveness of the enhancement proposed. We consider as benchmark problems the elementary shortest path problems arising as pricing subproblems in branch-and-price algorithms for the capacitated vehicle routing problem, the vehicle routing problem with distribution and collection and the capacitated vehicle routing problem with time windows.     

On the analysis of optimization problems in arc-dependent networks

This paper is concerned with the design and analysis of algorithms for optimization problems in arc-dependent networks. A network is said to be arc-dependent if the cost of an arc $a$ depends upon the arc taken to enter $a$. These networks are fundamentally different from traditional networks in which the cost associated with an arc is a fixed constant and part of the input. We first study the arc-dependent shortest path (ADSP) problem, which is also known as the suffix-1 path-dependent shortest path problem in the literature. This problem has a polynomial time solution if the shortest paths are not required to be simple. The ADSP problem finds applications in a number of domains, including highway engineering, turn penalties and prohibitions, and fare rebates. In this paper, we are interested in the ADSP problem when restricted to simple paths. We call this restricted version the simple arc-dependent shortest path (SADSP) problem. We show that the SADSP problem is NP-complete. We present inapproximability results and an exact exponential algorithm for this problem. We also extend our results for the longest path problem in arc-dependent networks. Additionally, we explore the problem of detecting negative cycles in arc-dependent networks and discuss its computational complexity. Our results include variants of the negative cycle detection problem such as longest, shortest, heaviest, and lightest negative simple cycles.²     

Algorithms for the quickest path problem and the reliable quickest path problem

The quickest path problem consists of finding a path in a directed network to transmit a given amount of items from an origin node to a destination node with minimal transmission time, when the transmission time depends on both the traversal times of the arcs, or lead time, and the rates of flow along arcs, or capacity. In telecommunications networks, arcs often also have an associated operational probability of the transmission being fault free. The reliability of a path is defined as the product of the operational probabilities of its arcs. The reliability as well as the transmission time are of interest. In this paper, algorithms are proposed to solve the quickest path problem as well as the problem of identifying the quickest path whose reliability is not lower than a given threshold. The algorithms rely on both the properties of a network which turns the computation of a quickest path into the computation of a shortest path and the fact that the reliability of a path can be evaluated through the reliability of the ordered sequence of its arcs. Other constraints on resources consumed, on the number of arcs of the path, etc. can also be managed with the same algorithms.     

Locating Active Sensors on Traffic Networks

Sensors are used to monitor traffic in networks. For example, in transportation networks, they may be used to measure traffic volumes on given arcs and paths of the network. This paper refers to an active sensor when it reads identifications of vehicles, including their routes in the network, that the vehicles actively provide when they use the network. On the other hand, the conventional inductance loop detectors are passive sensors that mostly count vehicles at points in a network to obtain traffic volumes (e.g., vehicles per hour) on a lane or road of the network.This paper introduces a new set of network location problems that determine where to locate active sensors in order to monitor or manage particular classes of identified traffic streams. In particular, it focuses on the development of two generic locational decision models for active sensors, which seek to answer these questions: (1) “How many and where should such sensors be located to obtain sufficient information on flow volumes on specified paths?”, and (2) “Given that the traffic management planners have already located count detectors on some network arcs, how many and where should active sensors be located to get the maximum information on flow volumes on specified paths?”The problem is formulated and analyzed for three different scenarios depending on whether there are already count detectors on arcs and if so, whether all the arcs or a fraction of them have them. Location of an active sensor results in a set of linear equations in path flow variables, whose solution provide the path flows. The general problem, which is related to the set-covering problem, is shown to be NP-Hard, but special cases are devised, where an arc may carry only two routes, that are shown to be polynomially solvable. New graph theoretic models and theorems are obtained for the latter cases, including the introduction of the generalized edge-covering by nodes problem on the path intersection graph for these special cases. An exact algorithm for the special cases and an approximate one for the general case are presented.     

Shortest paths in networks with vector weights

For a directed network in which vector weights are assigned to arcs, the Pareto analog to the shortest path problem is analyzed. An algorithm is presented for obtaining all Pareto shortest paths from a specified node to every other node.     

An implementation of linear and nonlinear multicommodity network flows

This work presents a new code for solving the multicommodity network flow problem with a linear or nonlinear objective function considering additional linear side constraints that link arcs of the same or different commodities. For the multicommodity network flow problem through primal partitioning the code implements a specialization of Murtagh and Saunders' strategy of dividing the set of variables into basic, nonbasic and superbasic. Several tests are reported, using random problems obtained from different network generators and real problems arising from the fields of long and short-term hydrothermal scheduling of electricity generation and traffic assignment, with sizes of up to 150000 variables and 45 000 constraints The performance of the code developed is compared to that of alternative methodologies for solving the same problems: a general purpose linear and nonlinear constrained optimization code, a specialised linear multicommodity network flow code and a primal-dual interior point code.     

Shortest path games

We study cooperative games that arise from the problem of finding shortest paths from a specified source to all other nodes in a network. Such networks model, among other things, efficient development of a commuter rail system for a growing metropolitan area. We motivate and define these games and provide reasonable conditions for the corresponding rail application. We show that the core of a shortest path game is nonempty and satisfies the given conditions, but that the Shapley value for these games may lie outside the core. However, we show that the shortest path game is convex for the special case of tree networks, and we provide a simple, polynomial time formula for the Shapley value in this case. In addition, we extend our tree results to the case where users of the network travel to nodes other than the source. Finally, we provide a necessary and sufficient condition for shortest paths to remain optimal in dynamic shortest path games, where nodes are added to the network sequentially over time.     

An extended shortest path problem: A data envelopment analysis approach

A special and important network structured linear programming problem is the shortest path problem. Classical shortest path problems assume that there are unit of shipping cost or profit along an arc. In many real occasions, various attributes (various costs and profits) are usually considered in a shortest path problem. Because of the frequent occurrence of such network structured problems, there is a need to develop an efficient procedure for handling these problems. This paper studies the shortest path problem in the case that multiple attributes are considered along the arcs. The concept of relative efficiency is defined for each path from initial node to final node. Then, an efficient path with the maximum efficiency is determined.     

An auction algorithm for the max-flow problem

We propose a new algorithm for the max-flow problem. It consists of a sequence of augmentations along paths constructed by an auction-like algorithm. These paths are not necessarily shortest, that is, they need not contain a minimum number of arcs. However, they can be found typically with much less computation than the shortest augmenting paths used by competing methods. Our algorithm outperforms these latter methods as well as state-of-the-art preflow-push algorithms by a very large margin in tests with standard randomly generated problems.This paper is a substantially revised version of Ref. 1.Many thanks are due to David Castanon and Paul Tseng for several helpful comments. The suggestions of the referees were also appreciated. David Castanon, Lakis Polymenakos, and Won-Jong Kim helped with some of the computational experimentation. This research was supported by NSF under Grant CCR-9103804.     

The first K shortest unique-arc walks in a traffic-light network

In this article, we present an algorithm to find the first K shortest unique-arc walks in a traffic-light network. Each node of the present network is associated with a repeated sequence of different windows to model operations of traffic signals and intersection movements. Unlike conventional simple or looping paths, we refer to the paths in this paper as unique-arc walks because they may include repeated nodes but will exclude repeated arcs. Using the heap structure, we develop an algorithm to find the first K shortest unique-arc walks in time O(Kr|V|³|A|), where |V| is the number of nodes, |A| is the number of arcs, and r represents the number of different windows associated with a node.     

灾后运输网络中的最短路修复合作博弈

在最短路修复合作博弈中,当灾后运输网络规模较大时,最优成本分摊问题难以直接求解。基于拉格朗日松弛理论,提出了一种最短路修复合作博弈成本分摊算法。该算法将最短路修复合作博弈分解为两个具有特殊结构的子博弈,进而利用两个子博弈的结构特性,可以{高效地}求解出二者的最优成本分摊,将这两个成本分摊相加,可以获得原博弈的一个近乎最优的稳定成本分摊。结果部分既包含运输网络的随机仿真,也包含玉树地震灾区的现实模拟,无论数据来源于仿真还是现实,该算法都能在短时间内为最短路修复合作博弈提供稳定的成本分摊方案。     

Finding K shortest looping paths with waiting time in a time–window network

A time-constrained shortest path problem is a shortest path problem including time constraints that are commonly modeled by the form of time windows. Finding K shortest paths are suitable for the problem associated with constraints that are difficult to define or optimize simultaneously. Depending on the types of constraints, these K paths are generally classified into either simple paths or looping paths. In the presence of time–window constraints, waiting time occurs but is largely ignored. Given a network with such constraints, the contribution of this paper is to develop a polynomial time algorithm that finds the first K shortest looping paths including waiting time. The time complexity of the algorithm is O(rK²|V₁|³), where r is the number of different windows of a node and |V₁| is the number of nodes in the original network.     

Network flow interdiction on planar graphs

The network flow interdiction problem asks to reduce the value of a maximum flow in a given network as much as possible by removing arcs and vertices of the network constrained to a fixed budget. Although the network flow interdiction problem is strongly NP-complete on general networks, pseudo-polynomial algorithms were found for planar networks with a single source and a single sink and without the possibility to remove vertices. In this work, we introduce pseudo-polynomial algorithms that overcome various restrictions of previous methods. In particular, we propose a planarity-preserving transformation that enables incorporation of vertex removals and vertex capacities in pseudo-polynomial interdiction algorithms for planar graphs. Additionally, a new approach is introduced that allows us to determine in pseudo-polynomial time the minimum interdiction budget needed to remove arcs and vertices of a given network such that the demands of the sink node cannot be completely satisfied anymore. The algorithm works on planar networks with multiple sources and sinks satisfying that the sum of the supplies at the sources equals the sum of the demands at the sinks. A simple extension of the proposed method allows us to broaden its applicability to solve network flow interdiction problems on planar networks with a single source and sink having no restrictions on the demand and supply. The proposed method can therefore solve a wider class of flow interdiction problems in pseudo-polynomial time than previous pseudo-polynomial algorithms and is the first pseudo-polynomial algorithm that can solve non-trivial planar flow interdiction problems with multiple sources and sinks. Furthermore, we show that the k-densest subgraph problem on planar graphs can be reduced to a network flow interdiction problem on a planar graph with multiple sources and sinks and polynomially bounded input numbers.     

Multiobjective design of survivable IP networks

Modern communication networks often use Internet Protocol routing and the intra-domain protocol OSPF (Open Shortest Path First). The routers in such a network calculate the shortest path to each destination and send the traffic on these paths, using load balancing. The issue of survivability, i.e. the question of how much traffic the network will be able to accommodate if components fail, is increasingly important. We consider the problem of designing a survivable IP network, which also requires determining the routing of the traffic. This is done by choosing the weights used for the shortest path calculations.     

Shortest Paths with Shortest Detours

This paper is concerned with a biobjective routing problem, called the shortest path with shortest detour problem, in which the length of a route is minimized as one criterion and, as second, the maximal length of a detour route if the chosen route is blocked is minimized. Furthermore, the relation to robust optimization is pointed out, and we present a new polynomial time algorithm, which computes a minimal complete set of efficient paths for the shortest path with shortest detour problem. Moreover, we show that the number of nondominated points is bounded by the number of arcs in the graph.     

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.