We consider the problem of finding shortest paths in a graph with independent randomly distributed edge lengths. Our goal is to maximize the probability that the path length does not exceed a given threshold value (deadline). We give a surprising exact n Θ(log n) algorithm for the case of normally distributed edge lengths, which is based on quasi-convex maximization. We then prove average and smoothed polynomial bounds for this algorithm, which also translate to average and smoothed bounds for the parametric shortest path problem, and extend to a more general non-convex optimization setting. We also consider a number other edge length distributions, giving a range of exact and approximation schemes.

The K shortest paths problem has been extensively studied for many years. Efficient methods have been devised, and many practical applications are known. Shortest hyperpath models have been proposed for several problems in different areas, for example in relation with routing in dynamic networks. However, the K shortest hyperpaths problem has not yet been investigated. In this paper we present procedures for finding the K shortest hyperpaths in a directed hypergraph. This is done by extending existing algorithms for K shortest loopless paths. Computational experiments on the proposed procedures are performed, and applications in transportation, planning and combinatorial optimization are discussed.

We present new complexity results and efficient algorithms for optimal route planning in the presence of uncertainty. We employ a decision theoretic framework for defining the optimal route: for a given source S and destination T in the graph, we seek an ST-path of lowest expected cost where the edge travel times are random variables and the cost is a nonlinear function of total travel time. Although this is a natural model for route-planning on real-world road networks, results are sparse due to the analytic difficulty of finding closed form expressions for the expected cost (Fan, Kalaba & Moore), as well as the computational/combinatorial difficulty of efficiently finding an optimal path which minimizes the expected cost. We identify a family of appropriate cost models and travel time distributions that are closed under convolution and physically valid. We obtain hardness results for routing problems with a given start time and cost functions with a global minimum, in a variety of deterministic and stochastic settings. In general the global cost is not separable into edge costs, precluding classic shortest-path approaches. However, using partial minimization techniques, we exhibit an efficient solution via dynamic programming with low polynomial complexity.

This paper examines the value of real-time traffic information to optimal vehicle routing in a nonstationary stochastic network. We present a systematic approach to aid in the implementation of transportation systems integrated with real time information technology. We develop decisionmaking procedures for determining the optimal driver attendance time, optimal departure times, and optimal routing policies under stochastically changing traffic flows based on a Markov decision process formulation. With a numerical study carried out on an urban road network in Southeast Michigan, we demonstrate significant advantages when using this information in terms of total costs savings and vehicle usage reduction while satisfying or improving service levels for just-in-time delivery.

The Shortest Path with Recourse Problem involves finding the shortest expected-length paths in a directed network each of whose arcs have stochastic traversal lengths (or delays) that become known only upon arrival at the tail of that arc. The traveler starts at a given source node, and makes routing decisions at each node in such a way that the expected distance to a given sink node is minimized. We develop an extension of Dijkstra’s algorithm to solve the version of the problem where arclengths are nonnegative and reset after each arc traversal. All known no-reset versions of the problem are NP-hard. We make a partial extension to the case where negative arclengths are present.

Abstract. The objective is to minimize expected travel time from any origin to a specific destination in a congestible network with correlated link costs. Each link is assumed to be in one of two possible conditions. Conditional probability density functions for link travel times are assumed known for each condition. Conditions over the traversed links are taken into account for determining the optimal routing strategy for the remaining trip. This problem is treated as a multi-stage adaptive feedback control process. Each stage is described by the physical state (the location of the current decision point) and the information state (the service level of the previously traversed links). Proof of existence and uniqueness of the solution to the basic dynamic programming equations and a solution procedure are provided. Key Words. Shortest path, stochastic networks, dynamic programming, adaptive feedback control, correlated link costs.

Routing vehicles based on real-time traffic information has been shown to significantly reduce travel time, and hence cost, in high-volume traffic situations. However, taking real-time traffic data and transforming them into optimal route decisions is a computational challenge. We model the dynamic route determination problem as a Markov decision process (MDP) and present procedures for identifying traffic data having no decision-making value. Such identification can be used to reduce the state space of the MDP, improving its computational tractability. This reduction can be achieved by a two-step process. The first is an a priori reduction that may be performed using a stationary, deterministic network with upper and lower bounds on the cost functions before the trip begins. The second part of the process dynamically reduces the state space further on the nonstationary stochastic road network as the trip optimally progresses. We demonstrate the significant computational advantages of the introduced methods based on an actual road network in southeast Michigan.

This dissertation was presented by Tomas Singliar. This thesis brings a collection of novel models and methods that result from a new look at practical problems in transportation through the prism of newly available sensor data. There are four main contributions: First, we design a generative probabilistic graphical model to describe multivariate continuous densities such as observed traffic patterns. The model implements a multivariate normal distribution with covariance constrained in a natural way, using a number of parameters that is only linear (as opposed to quadratic) in the dimensionality of the data. This means that learning these models requires less data. The primary use for such a model is to support inferences, for instance, of data missing due to sensor malfunctions. Second, we build a model of traffic flow inspired by macroscopic flow models. Unlike traditional such models, our model deals with uncertainty of measurement and unobservability of certain important quantities and incorporates on-the-fly observations more easily. Because the model does not admit efficient exact inference, we develop a particle filter. The model delivers better medium- and long- term predictions than general-purpose time series models. Moreover, having a predictive distribution of traffic state enables the application of powerful decision-making machinery to the traffic domain. Third, two new optimization algorithms for the common task of vehicle routing are designed, using the tra±c °ow model as their probabilistic underpinning. Their bene¯ts include suitability to highly volatile environments and the fact that optimization criteria other than the classical minimal expected time are easily incorporated. Finally, we present a new method for detecting accidents and other adverse events. Data collected from highways enables us to bring supervised learning approaches to incident detection. We show that a support vector machine learner can outperform manually calibrated solutions. A major hurdle to performance of supervised learners is the quality of data which contains systematic biases varying from site to site. We build a dynamic Bayesian network framework that learns and rectifies these biases, leading to improved supervised detector performance with little need for manually tagged data. The realignment method applies generally to virtually all forms of labeled sequential data.

Abstract. We consider a stochastic routing model in which the goal is to find the optimal route that incorporates a measure of risk. The settings such as task planning (where the time to execute tasks is uncertain), etc. The stochasticity is specified in terms of arbitrary edge length distributions with given mean and variance values in a graph. The objective function is a positive linear combination of the mean and standard deviation of the route. Both the nonconvex objective and exponentially sized feasible set of available routes present a challenging optimization problem for which no efficient algorithms are known. In this paper we evaluate the practical performance of algorithms and heuristic approaches which show very promising results in terms of both running time and solution accuracy. 1

Transportation Center. DISCLAIMERS The contents of this report reflect the views of the authors, who are responsible for the facts and the accuracy of the information presented herein. This document is disseminated under the sponsorship of the Department of Transportation University Transportation Centers Program, in the interest of information exchange. The U.S. Government assumes no liability for the contents or use thereof. The opinions, findings and conclusions expressed in this publication are those of the authors