Computing shortest paths is one of the most researched topics in algorithm engineering. Currently available algorithms compute shortest paths in mere fractions of a second on continental sized road networks. In the presence of unreliability, however, current algorithms fail to achieve results as impressive as for the static setting. In contrast to speed-up techniques for static route planning, current implementations for the stochastic on-time arrival problem require the computationally expensive step of solving convolution products. Running times can reach hours when considering large scale networks. We present a novel approach to reduce this immense computational effort of stochastic routing based on existing techniques for alternative routes. In an extensive experimental study, we show that the process of stochastic route planning can be speed-up immensely, without sacrificing much in terms of accuracy.
One approach to unreliable travel times is to take into account the inherent variability in the travel time distributions and compute robust routing strategies. A commonly used method in this setting is to maximize the probability of reaching the destination within a prespecified time budget. There are two different approaches that have been developed to solve this problem. The first approach is to find the optimal path that maximizes the probability of on-time arrival and is known as the shortest path with on-time arrival reliability (SPOTAR) problem. Nie and Wu @cite_5 proposed a general solution to the problem based on first-order stochastic dominance of paths, but the solution has worst-case exponential running times. They also present a pseudo-polynomial algorithm that gives an approximate solution to the problem and performs well in practice. @cite_7 showed how to solve the problem in @math time when the arc travel time distributions are Gaussian.
@cite_5 @cite_7
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This paper studies the problem of finding a priori shortest paths to guarantee a given likelihood of arriving on-time in a stochastic network. Such "reliable" paths help travelers better plan their trips to prepare for the risk of running late in the face of stochastic travel times. Optimal solutions to the problem can be obtained from local-reliable paths, which are a set of non-dominated paths under first-order stochastic dominance. We show that Bellman's principle of optimality can be applied to construct local-reliable paths. Acyclicity of local-reliable paths is established and used for proving finite convergence of solution procedures. The connection between the a priori path problem and the corresponding adaptive routing problem is also revealed. A label-correcting algorithm is proposed and its complexity is analyzed. A pseudo-polynomial approximation is proposed based on extreme-dominance. An extension that allows travel time distribution functions to vary over time is also discussed. We show that the time-dependent problem is decomposable with respect to arrival times and therefore can be solved as easily as its static counterpart. Numerical results are provided using typical transportation networks. We consider the problem of computing shortest paths through a dynamic network – a network with time-varying characteristics, such as arc travel times and costs, which are known for all values of time. Many types of networks, most notably transportation networks, exhibit such predictable dynamic behavior over the course of time. Dynamic shortest path problems are currently solved in practice by algorithms which operate within a discrete-time framework. In this thesis, we introduce a new set of algorithms for computing shortest paths in continuous-time dynamic networks, and demonstrate for the first time in the literature the feasibility and the advantages of solving dynamic shortest path problems in continuous time. We assume that all time-dependent network data functions are given as piece-wise linear functions of time, a representation capable of easily modeling most common dynamic problems. Additionally, this form of representation and the solution algorithms developed in this thesis are well suited for many augmented static problems such as time-constrained minimum-cost shortest path problems and shortest path problems with time windows. We discuss the classification, formulation, and mathematical properties of all common variants of the continuous-time dynamic shortest path problem. Two classes of solution algorithms are introduced, both of which are shown to solve all variants of the problem. In problems where arc travel time functions exhibit First-In-First-Out (FIFO) behavior, we show that these algorithms have polynomial running time; although the general problem is NP-hard, we argue that the average-case running time for many common problems should be quite reasonable. Computational results are given which support the theoretical analysis of these algorithms, and which provide a comparison with existing discrete-time algorithms; in most cases, continuous-time approaches are shown to be much more efficient, both in running time and storage requirements, than their discretetime counterparts. Finally, in order to further reduce computation time, we introduce parallel algorithms, and hybrid continuous-discrete approximation algorithms which exploit favorable characteristics of algorithms from both domains. Thesis Supervisor: Ismail Chabini Title: Assistant Professor, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology Abstract The shortest path problem is of great importance to the real world in such areas as transportation, network design, telecommunication, etc. The deterministic version of the problem is easily solved. However, in the real world, uncertainty is frequently encountered and must be dealt with. For instance, in a transportation network, the path that has the least travel time when no traffic is present might be prone to accidents and traffic jams, and thus it may lead to a drastic increase in travel time during rush hours. In situations with significant uncertainty, the deterministic approach can be far from sufficient. New and appropriate criteria and models to handle uncertainties along with efficient solution techniques are in need. In this paper, we take a minimax approach to measure the performance under uncertainties. Under our measure, one path is superior to all others only if its worst case performance is better than that of the others. The complexity of the defined models, exact and heuristic solution methods are explored and presented. Analysis is done to examine the performance of the heuristic method. The shortest path (SP) problem in a network with nonnegative arc lengths can be solved easily by Dijkstra's labeling algorithm in polynomial time. In the case of significant uncertainty of the arc lengths, a robustness approach is more appropriate. In this paper, we study the SP problem under arc length uncertainties. A scenario approach is adopted to characterize uncertainties. Two robustness criteria are specified: the absolute robust criterion and the robust deviation criterion. We show that under both criteria the robust SP problem is NP-complete even for the much more restricted layered networks of width 2, and with only 2 scenarios. A pseudo-polynomial algorithm is devised to solve the robust SP problem in general networks under bounded number of scenarios. Also presented is a more efficient algorithm for layered networks. However, in the case of unlimited number of scenarios, we show that the robust SP problem is strongly NP-hard. A simple heuristic for finding a good robust shortest path is provided, and the worst case performance is analyzed. We address the problem of choosing the best paths among a set of candidate paths between the same origin–destination pair. This functionality is used extensively when constructing origin–destination matrices in logistics and flex transportation. Because the cost of a path, e.g., travel time, varies over time and is uncertain, there is generally no single best path. We partition time into intervals and represent the cost of a path during an interval as a random variable, resulting in an uncertain time series for each path. When facing uncertainties, users generally have different risk preferences, e.g., risk-loving or risk-averse, and thus prefer different paths. We develop techniques that, for each time interval, are able to find paths with non-dominated lowest costs while taking the users’ risk preferences into account. We represent risk by means of utility function categories and show how the use of first-order and two kinds of second-order stochastic dominance relationships among random variables makes it possible to find all paths with non-dominated lowest costs. We report on empirical studies with large uncertain time series collections derived from a 2-year GPS data set. The study offers insight into the performance of the proposed techniques, and it indicates that the best techniques combine to offer an efficient and robust solution. We consider the stochastic on-time arrival (SOTA) problem of finding the optimal routing strategy for reaching a given destination within a pre-specified time budget and provide the first results on using preprocessing techniques for speeding up the query time. We start by identifying some properties of the SOTA problem that limit the types of preprocessing techniques that can be used in this setting, and then define the stochastic variants of two deterministic shortest path preprocessing techniques that can be adapted to the SOTA problem, namely reach and arc-flags. We present the preprocessing and query algorithms for each technique, and also present an extension to the standard reach based preprocessing method that provides additional pruning. Finally, we explain the limitations of this approach due to the inefficiency of the preprocessing phase and present a fast heuristic preprocessing scheme. Numerical results for San Francisco, Luxembourg and a synthetic road network show up to an order of magnitude improvement in the query time for short queries, with even larger gains expected for longer queries. We study a new robust path problem, the Online Replacement Path problem (ORP). Consider the problem of routing a physical package through a faulty network G = (V,E) from a source s ∈ V to a destination t ∈ V as quickly as possible. An adversary, whose objective is to maximize the latter routing time, can choose to remove a single edge in the network. In one setup, the identity of the edge is revealed to the routing mechanism (RM) while the package is in s. In this setup the best strategy is to route the package along the shortest path in the remaining network. The payoff maximization problem for the adversary becomes the Most Vital Arc problem (MVA), which amounts to choosing the edge in the network whose removal results in a maximal increase of the s-t distance. However, the assumption that the RM is informed about the failed edge when standing at s is unrealistic in many applications, in which failures occur online, and, in particular, after the routing has started. We therefore consider the setup in which the adversary can reveal the identity of the failed edge just before the RM attempts to use this edge, thus forcing it to use a different route to t, starting from the current node. The problem of choosing the nominal path minimizing the worst case arrival time at t in this setup is ORP. We show that ORP can be solved in polynomial time and study other models naturally providing middle grounds between MVA and ORP. Our results show that ORP comprises a highly flexible and tractable framework for dealing with robustness issues in the design of RM-s. We discuss the problem of computing shortest anisotropic paths on terrains. Anisotropic path costs take into account the length of the path traveled, possibly weighted, and the direction of travel along the faces of the terrain. Considering faces to be weighted has added realism to the study of (pure) Euclidean shortest paths. Parameters such as the varied nature of the terrain, friction, or slope of each face, can be captured via face weights. Anisotropic paths add further realism by taking into consideration the direction of travel on each face thereby e.g., eliminating paths that are too steep for vehicles to travel and preventing the vehicles from turning over. Prior to this work an O(nn) time algorithm had been presented for computing anisotropic paths. Here we present the first polynomial time approximation algorithm for computing shortest anisotropic paths. Our algorithm is simple to implement and allows for the computation of shortest anisotropic paths within a desired accuracy. Our result addresses the corresponding problem posed in [12]. Maximizing information gathered within a budget is a relevant problem for information gathering tasks for robots with cost or operating time constraints. This problem is also known as the informative path planning (IPP) problem or correlated orienteering. It can be formalized as that of finding budgeted routes in a graph such that the reward collected by the route is maximized, where the reward at nodes can be dependent. Unfortunately, the problem is NP-Hard and the state of the art methods are too slow to even present an approximate solution online. Here we present Randomized Anytime Orienteering (RAOr) algorithm that provides near optimal solutions while demonstrably converging to an efficient solution in runtimes that allows the solver to be run online. The key idea of our approach is to pose orienteering as a combination of a Constraint Satisfaction Problem and a Traveling Salesman Problem. This formulation allows us to restrict the search space to routes that incur minimum distance to visit a set of selected nodes, and rapidly search this space using random sampling. The paper provides the analysis of asymptotic near-optimality, convergence rates for RAOr algorithms, and present strategies to improve anytime performance of the algorithm. Our experimental results suggest an improvement by an order of magnitude over the state of the art methods in relevant simulation and in real world scenarios. This paper resolves the complexity status of the well-known Traveling Repairman Problem on a line (Line-TRP) with general processing times at the request locations and deadline restrictions. It has long remained an open research question whether an exact solution procedure with pseudo-polynomial running time can be developed for this version of the Traveling Repairman Problem that was known to be at least binary NP-hard. The presented proof of strong NP-completeness of the problem is provided by a reduction from 3-PARTITION. Since recent literature provides significant new results for further variants of the Line-TRP and the Line-TSP, a brief updated overview of the complexity status of the different variants is given. Another major contribution is that a practically applicable exact best-first search Branch&Bound approach that optimally solves instances of real-world size in reasonable time is proposed. By applying sophisticated dominance rules as well as lower bounds, the number of enumerated partial solutions is kept limited. The efficiency of the new approach and the applied instruments is validated by a computational study. Several polynomial time approximation algorithms for some @math -complete routing problems are presented, and the worst-case ratios of the cost of the obtained route to that of an optimal are determined. A mixed-strategy heuristic with a bound of 9 5 is presented for the stacker-crane problem (a modified traveling salesman problem). A tour-splitting heuristic is given for k-person variants of the traveling salesman problem, the Chinese postman problem, and the stacker-crane problem, for which a minimax solution is sought. This heuristic has a bound of @math , where e is the bound for the corresponding 1-person algorithm. We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum-cost spanning tree, minimum-weight perfect matching, traveling salesman and Steiner tree problems. Our technique produces approximation algorithms that run in @math time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximation algorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of @math time compares favorably with the best strongly polynomial exact algorithms running in @math time for dense graphs. A similar result is obtained for the 2-matching problem and its variants. We also derive the first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413--420]. In this paper, we propose a citywide and real-time model for estimating the travel time of any path (represented as a sequence of connected road segments) in real time in a city, based on the GPS trajectories of vehicles received in current time slots and over a period of history as well as map data sources. Though this is a strategically important task in many traffic monitoring and routing systems, the problem has not been well solved yet given the following three challenges. The first is the data sparsity problem, i.e., many road segments may not be traveled by any GPS-equipped vehicles in present time slot. In most cases, we cannot find a trajectory exactly traversing a query path either. Second, for the fragment of a path with trajectories, they are multiple ways of using (or combining) the trajectories to estimate the corresponding travel time. Finding an optimal combination is a challenging problem, subject to a tradeoff between the length of a path and the number of trajectories traversing the path (i.e., support). Third, we need to instantly answer users' queries which may occur in any part of a given city. This calls for an efficient, scalable and effective solution that can enable a citywide and real-time travel time estimation. To address these challenges, we model different drivers' travel times on different road segments in different time slots with a three dimension tensor. Combined with geospatial, temporal and historical contexts learned from trajectories and map data, we fill in the tensor's missing values through a context-aware tensor decomposition approach. We then devise and prove an object function to model the aforementioned tradeoff, with which we find the most optimal concatenation of trajectories for an estimate through a dynamic programming solution. In addition, we propose using frequent trajectory patterns (mined from historical trajectories) to scale down the candidates of concatenation and a suffix-tree-based index to manage the trajectories received in the present time slot. We evaluate our method based on extensive experiments, using GPS trajectories generated by more than 32,000 taxis over a period of two months. The results demonstrate the effectiveness, efficiency and scalability of our method beyond baseline approaches. In this paper, we give the first constant-factor approximation algorithm for the rooted Orienteering problem, as well as a new problem that we call the Discounted-Reward traveling salesman problem (TSP), motivated by robot navigation. In both problems, we are given a graph with lengths on edges and rewards on nodes, and a start node @math . In the Orienteering problem, the goal is to find a path starting at @math that maximizes the reward collected, subject to a hard limit on the total length of the path. In the Discounted-Reward TSP, instead of a length limit we are given a discount factor @math , and the goal is to maximize the total discounted reward collected, where the reward for a node reached at time @math is discounted by @math . This problem is motivated by an approximation to a planning problem in the Markov decision process (MDP) framework under the commonly employed infinite horizon discounted reward optimality criterion. The approximation arises from a need to deal with exponentially large state spaces that emerge when trying to model one-time events and nonrepeatable rewards (such as for package deliveries). We also consider tree and multiple-path variants of these problems and provide approximations for those as well. Although the unrooted Orienteering problem, where there is no fixed start node @math , has been known to be approximable using algorithms for related problems such as @math -TSP (in which the amount of reward to be collected is fixed and the total length is approximately minimized), ours is the first to approximate the rooted question, solving an open problem in [E. M. Arkin, J. S. B. Mitchell, and G. Narasimhan, Proceedings of the @math th ACM Symposium on Computational Geometry, 1998, pp. 307-316] and [B. Awerbuch, Y. Azar, A. Blum, and S. Vempala, SIAM J. Comput., 28 (1998), pp. 254-262]. We complement our approximation result for Orienteering by showing that the problem is APX-hard. The development of intelligent transportation systems (ITS) and the resulting need for the solution of a variety of dynamic traffic network models and management problems require faster-than-real-time computation of shortest path problems in dynamic networks. Recently, a sequential algorithm was developed to compute shortest paths in discrete time dynamic networks from all nodes and all departure times to one destination node. The algorithm is known as algorithm DOT and has an optimal worst-case running-time complexity. This implies that no algorithm with a better worst-case computational complexity can be discovered. Consequently, in order to derive algorithms to solve all-to-one shortest path problems in dynamic networks, one would need to explore avenues other than the design of sequential solution algorithms only. The use of commercially-available high-performance computing platforms to develop parallel implementations of sequential algorithms is an example of such avenue. This paper reports on the design, implementation, and computational testing of parallel dynamic shortest path algorithms. We develop two shared-memory and two message-passing dynamic shortest path algorithm implementations, which are derived from algorithm DOT using the following parallelization strategies: decomposition by destination and decomposition by transportation network topology. The algorithms are coded using two types of parallel computing environments: a message-passing environment based on the parallel virtual machine (PVM) library and a multi-threading environment based on the SUN Microsystems Multi-Threads (MT) library. We also develop a time-based parallel version of algorithm DOT for the case of minimum time paths in FIFO networks, and a theoretical parallelization of algorithm DOT on an ‘ideal’ theoretical parallel machine. Performances of the implementations are analyzed and evaluated using large transportation networks, and two types of parallel computing platforms: a distributed network of Unix workstations and a SUN shared-memory machine containing eight processors. Satisfactory speed-ups in the running time of sequential algorithms are achieved, in particular for shared-memory machines. Numerical results indicate that shared-memory computers constitute the most appropriate type of parallel computing platforms for the computation of dynamic shortest paths for real-time ITS applications. This paper derives a simple fast algorithm for computing minimal-revenue tolls in a single-origin network. It assumes that trips have the same value of time, that they make user-optimizing path choices, and they have multiple destinations--but come from the same origin. The algorithm finds tolls that induce a traffic pattern minimizing average time per trip at a minimal average toll per trip. Formally, let X be the set of all feasible traffic assignments and assume all trips have the same value of time [alpha]>0; then the algorithm inputs a network with the system-optimizing flow xo vector and associated arc times t(xo). It outputs an optimal arc toll vector c*[greater-or-equal, slanted]0 such that for x[set membership, variant]X:([alpha]t(xo)+c*)T(x-xo)[greater-or-equal, slanted]0 Â forÂ x[set membership, variant]X (user Â optimal) That is, xo[set membership, variant]X becomes user-optimal, too. Compared to marginal-cost tolls, which would also produce this same effect, the min-revenue tolls c* possess important practical advantages. Perforce, they provide the same system-optimal network usage, but at a much cheaper price to the traveler--remarkably, they assure at least one toll-free path to every destination. In addition, min-revenue tolls have admirable stability: even large increases in travel demand can leave the solution c* virtually unchanged--an important feature since traffic volume can change continuously but tolls cannot. Finally, they provide guidance regarding potential network improvements. The algorithm presented is very efficient, easily able to solve networks with tens of thousands of nodes. A greedy potentials calculator, its expected running time for an n-node road network is a speedy O(n), while its worst case time for any network is O(m), where m is the number of arcs. The present study investigates the cost concerns of distribution centers and formulates a vehicle routing problem with time window constraints accordingly. Based on the embedded structure of the original problem, a decomposition technique is employed to decompose the original problems to a clustering problem (main problem) and a set of traveling salesman problems (sub-problems) with time window constraints. This decomposition not only reduces the problem size but also enable the use of simpler solution procedures. A genetic algorithm is developed to solve the clustering problem, while a simple heuristic algorithm is formulated to solve the set of traveling salesman problems. The solution of the original problem is obtained through iterative interactions between the main problem and the set of sub-problems. The performance of the proposed approach is compared with the well-known insertion method and a manual scheduling of a distribution center. We consider the problem of maximizing the revenue raised from tolls set on the arcs of a transportation network, under the constraint that users are assigned to toll-compatible shortest paths. We first prove that this problem is strongly NP-hard. We then provide a polynomial time algorithm with a worst-case precision guarantee of @math , where mT denotes the number of toll arcs. Finally, we show that the approximation is tight with respect to a natural relaxation by constructing a family of instances for which the relaxation gap is reached. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(1), 57–67 2005 The second of a two-part series, this paper derives an efficient solution to the minimal-revenue tolls problem. As introduced in Part I, this problem can be defined as follows: Assuming each trip uses only a path whose generalized cost is smallest, find a set of arc tolls that simultaneously minimizes both average travel time and out-of-pocket cost. As a point of departure, this paper first re-solves the single-origin problem of Part I, modeling it as a linear program. Then with a change of variable, it transforms the LP's dual into a simple longest-path problem on an acyclic network. The multiple-origin problem - where one toll for each arc applies to all origins - solves analogously. In this case, however, the dual becomes an elementary linear multi-commodity max-cost flow problem with an easy bundling constraint and infinite arc capacities. After a minor reformulation that simplifies the model's input to better accommodate output from common traffic assignment software, a solution algorithm is exemplified with a numerical example. Travel route planning is an important step for a tourist to prepare his her trip. As a common scenario, a tourist usually asks the following questions when he she is planning his her trip in an unfamiliar place: 1) Are there any travel route suggestions for a one-day or three-day trip in Beijing? 2) What is the most popular travel path within the Forbidden City? To facilitate a tourist's trip planning, in this paper, we target at solving the problem of automatic travel route planning. We propose to leverage existing travel clues recovered from 20 million geo-tagged photos collected from www.panoramio.com to suggest customized travel route plans according to users' preferences. As the footprints of tourists at memorable destinations, the geo-tagged photos could be naturally used to discover the travel paths within a destination (attractions landmarks) and travel routes between destinations. Based on the information discovered from geo-tagged photos, we can provide a customized trip plan for a tourist, i.e., the popular destinations to visit, the visiting order of destinations, the time arrangement in each destination, and the typical travel path within each destination. Users are also enabled to specify personal preference such as visiting location, visiting time season, travel duration, and destination style in an interactive manner to guide the system. Owning to 20 million geo-tagged photos and 200,000 travelogues, an online system has been developed to help users plan travel routes for over 30,000 attractions landmarks in more than 100 countries and territories. Experimental results show the intelligence and effectiveness of the proposed framework. We study the directed minimum latency problem: given an n-vertex asymmetric metric (V,d) with a root vertex ri¾? V, find a spanning path originating at rthat minimizes the sum of latencies at all vertices (the latency of any vertex vi¾? Vis the distance from rto valong the path). This problem has been well-studied on symmetric metrics, and the best known approximation guarantee is 3.59 [3]. For any @math O( n^ ^3 ) @math =O( n )$, which implies (for any fixed i¾?> 0) a polynomial time O(n1 2 + i¾?)-approximation algorithm for directed latency. In the special case of metrics induced by shortest-paths in an unweighted directed graph, we give an O(log2n) approximation algorithm. As a consequence, we also obtain an O(log2n) approximation algorithm for minimizing the weighted completion time in no-wait permutation flowshop scheduling. We note that even in unweighted directed graphs, the directed latency problem is at least as hard to approximate as the well-studied asymmetric traveling salesman problem, for which the best known approximation guarantee is O(logn).