Dynamic Network Flow Problem Research Articles

Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

Modeling Minimum Cost Network Flows With Port‐Hamiltonian Systems

AbstractWe give a short overview of advantages and drawbacks of the classical formulation of minimum cost network flow problems and solution techniques, to motivate a reformulation of classical static minimum cost network flow problems as optimal control problems constrained by port‐Hamiltonian systems (pHS). The first‐order optimality system for the port‐Hamiltonian system‐constrained optimal control problem is formally derived. Then we propose a gradient‐based algorithm to find optimal controls. The port‐Hamiltonian system formulation naturally conserves flow and supports a wide array of further modeling options as, for example, node reservoirs, flow dependent costs, leaking pipes (dissipation) and coupled sub‐networks (ports). They thus provide a versatile alternative to state‐of‐the art approaches towards dynamic network flow problems, which are often based on computationally costly time‐expanded networks. We argue that this opens the door for a plethora of modeling options and solution approaches for network flow problems.

Quickest Flow Algorithms with Time-Varying Attributes

In many real-world situations, there are numerous network optimization problems where the network attributes depend on time. In this paper, we consider single-source single-sink discrete-time dynamic network flow problems. We review some algorithms for the quickest flow problems in two environments (to the network attributes): time-invariant and time-variant. This paper mainly focuses on the existing algorithms for a later one. In literature, most of the authors have made their objectives to determine the earliest arrival time paths along which a given amount of flow can be sent in the minimum time. Evacuation is the most recent research area of network optimization, where quickest flow models allow the estimation of the minimum time required to bring a given number of evacuees to safety.

Dynamic network flow location models and algorithms for quickest evacuation planning

Dynamic network flow problems have wide applications in evacuation planning. From a given subset of arcs in a directed network, choosing the suitable arcs for facility location with a given objective is very important in the optimization of flow in emergency cases. Because of the decrease in capacity of an arc by placing a facility in it, there may be a reduction in the maximum flow or increase in the quickest time. In this work, we consider a problem of identifying the optimal facility locations so that the increase in the quickest time is minimum. Introducing the quickest FlowLoc problem, we give strongly polynomial time algorithms to solve the single facility case. Realizing NP-hardness of the multi-facility case, we develop a mixed integer programming formulation of it and propose two polynomial time heuristics for its solution. Because of the growing concerns of arc reversals in evacuation planning, we introduce the quickest ContraFlowLoc problem and present exact algorithms to solve the single-facility case and heuristics to solve the multi-facility case, with polynomial time complexity. The solutions thus obtained here are practically important, particularly in evacuation planning, to systematize traffic flow with facility allocation minimizing evacuation time.

Fairness and efficiency in pedestrian emergency evacuation: Modeling and simulation

The goal of this paper is to study different algorithms for planning a large scale pedestrian evacuation process. A pedestrian (walking) mode of evacuation is an efficient mode of transportation that is often overlooked in the literature. Fairness in the evacuation process is another aspect that is rarely studied. In this paper, we study efficiency and fairness of pedestrian mode of emergency evacuation. We model the evacuation problem as a dynamic network flow problem, which captures the time-dependency of the network attributes such as edge and vertex capacities and edge travel times. To evaluate the efficiency of the algorithms, we consider the clearance time of the network. To investigate fairness, we consider the Gini coefficient and Lorenz curve that are popular indexes in economy to measure wealth distribution in a society. We compute the Gini coefficient and Lorenz curve to measure the fairness in travel time distribution of the evacuees in the emergency evacuation process. We develop a new algorithm called PEPA (Pedestrian Evacuation Planning Algorithm) to find the minimum clearance time of the network. PEPA is compared to two commonly studied graph algorithms in the literature, shortest path and maximum flow algorithms. We compared the algorithms in terms of computation time of the algorithm, clearance time of the network and Gini coefficient. Our results show that PEPA beats both shortest path and maximum flow in terms of efficiency and fairness, although it is slightly slower in terms of computation time. To implement PEPA, shortest path and maximum flow algorithms, we develop a new software named PEMSS (Pedestrian Evacuation Modeler, Solver, and Simulator). PEMSS is written in the Java programming language and is capable of reading and showing maps in open street map XML format.

A new dynamic network flow algorithm using base state amendment model for emergency response

AbstractLarge‐scale earthquake disasters in recent years have caused huge damage to people's lives and property. Quick effective emergency response is the primary task post‐disaster. A dynamic network flow model can be used to obtain an effective emergency material rescue plan for the transportation network. Strong timeliness is a critical factor to emergency response. However, very few efficient algorithms were presented for most dynamic network flow problems. To solve this issue, this article introduces a novel low‐time‐complexity exact algorithm using the continuous‐time dynamic network flow (CTDNF) and the base state with amendment (BSA) model in the geographic information science field. Besides, this new CTDNF‐BSA algorithm is improved to solve more complex cases with dynamic capacity and transmit time in a dynamic transportation network. Both numerical and geographic experiments were conducted to assess the correctness, time and space performance of the presented CTDNF‐BSA algorithm. The experimental results demonstrated that the CTDNF‐BSA algorithm can attain the optimal solution with good computing time and space performance as compared with traditional algorithms. Geographic cases in dynamic situations for real emergency response illustrated that the CTDNF‐BSA algorithm can provide effective decision support for emergency material rescue planning in a dynamic transportation environment.

Clearance Time Estimation for Incorporating Evacuation Risk in Routing Strategies for Evacuation Operations

This study seeks to develop an approximate but efficient approach for estimating evacuation clearance time which is defined as the time required to evacuate the population of a location to areas of safety. Elsewhere, this estimate of the clearance time is used as input to infer evacuation risk for a location which reflects whether the population of that location can be safely evacuated before the disaster impacts it. As the computed evacuation risk is used in a real-time stage-based framework for evacuation operations, the approach for clearance time estimation needs to be computationally efficient while being capable of approximating traffic flow dynamics reasonably. To address these needs, a dynamic routing policy labeled the location-priority routing is proposed, and the associated clearance time estimation problem is formulated as a dynamic network flow problem. The location-priority is based on the lead time that a location has until it is impacted by the disaster, and designates that the population at a location with a shorter lead time has higher priority in using roadway capacity for evacuation. By combining the location-priority routing with the consideration of a super sink, the routing problem for the evacuation network is transformed into sequential single-origin single-destination dynamic routing problems. It avoids expensive iterative search processes, enabling computational efficiency for real-time evacuation operations. The solution approach approximately captures the evolution of traffic dynamics across the stages of the operational framework. Results from numerical experiments illustrate that the proposed approach can address the aforementioned needs related to clearance time estimation.

Optimal egress time calculation and path generation for large evacuation networks

Finding the optimal clearance time and deciding the path and schedule of evacuation for large networks have traditionally been computationally intensive. In this paper, we propose a new method for finding the solution for this dynamic network flow problem with considerably lower computation time. Using a three phase solution method, we provide solutions for required clearance time for complete evacuation, minimum number of evacuation paths required for evacuation in least possible time and the starting schedules on those paths. First, a lower bound on the clearance time is calculated using minimum cost dynamic network flow model on a modified network graph representing the transportation network. Next, a solution pool of feasible paths between all O-D pairs is generated. Using the input from the first two models, a flow assignment model is developed to select the best paths from the pool and assign flow and decide schedule for evacuation with lowest clearance time possible. All the proposed models are mixed integer linear programing models and formulation is done for System Optimum (SO) scenario where the emphasis is on complete network evacuation in minimum possible clearance time without any preset priority. We demonstrate that the model can handle large size networks with low computation time. A numerical example illustrates the applicability and effectiveness of the proposed approach for evacuation.

Algorithm for Intermodal Optimal Multidestination Tour with Dynamic Travel Times

This paper presents an efficient algorithm that finds the intermodal optimal tour (origin to origin) in a time-dependent transportation network while the algorithm implicitly solves the park-and-ride facility choice problem with the inherent park-and-ride constraints for a traveler with a sequence of destinations to visit. To solve the problem, a network expansion technique that captures the constraints of park-and-ride behavior in the model and that transforms the park-and-ride choice problem into a dynamic network flow problem is introduced. An efficient iterative labeling algorithm that finds the optimal intermodal tour to serve the sequence of activities is also introduced. Multisource shortest-path runs are used in the iterative labeling algorithm to find the optimal tour with several intermediate destinations in an efficient manner. The performance of the algorithm is compared with the performance of existing approaches, and improvement is indicated. The solution method proposed benefits from the advantages of Dijkstra's shortest-path algorithm, which is made possible by (a) a nontrivial transformation of the original problem into a dynamic network flow problem and (b) an innovative use of a multisource shortest path in the context of origin–destination choice. The solution algorithm integrates time-dependent auto and transit shortest-path algorithms to find the optimal tour. The algorithm is implemented, coded, and tested on a real network, and the results are promising.

Determining the Optimal Flows in Zero-Time Dynamic Networks

Here we are dealing with minimum cost flow problem on dynamic network flows with zero transit times and a new arc capacity, horizon capacity, which denotes an upper bound on the total flow traversing through on an arc during a pre-specified time horizon T. We develop a simple approach based on mathematical modelling attributes to solve the min-cost dynamic network flow problem where arc capacities and costs are time varying, and horizon capacities are considered. The basis of the method is simple and relies on the appropriate defining of polyhedrons, and in contrast to the other usual algorithms that use the notion of time expanded network, this method runs directly on the original network.

On solving continuous-time dynamic network flows

Temporal dynamics is a crucial feature of network flow problems occurring in many practical applications. Important characteristics of real-world networks such as arc capacities, transit times, transit and storage costs, demands and supplies etc. are subject to fluctuations over time. Consequently, also flow on arcs can change over time which leads to so-called dynamic network flows. While time is a continuous entity by nature, discrete-time models are often used for modeling dynamic network flows as the resulting problems are in general much easier to handle computationally. In this paper, we study a general class of dynamic network flow problems in the continuous-time model, where the input functions are assumed to be piecewise linear or piecewise constant. We give two discrete approximations of the problem by dividing the considered time range into intervals where all parameters are constant or linear. We then present two algorithms that compute, or at least converge to optimum solutions. Finally, we give an empirical analysis of the performance of both algorithms.

A basic mathematical model for evacuation problems in urban areas

Real life situations like floods, hurricanes or chemical accidents may cause the evacuation of a certain area to rescue the affected population. To enable a fast and a safe evacuation a basic mixed-integer evacuation model has been developed that provides a reorganization of the traffic routing of a certain area for the case of an evacuation. This basic problem of evacuation minimizes the evacuation-time while prohibiting conflicts within intersections. Our evacuation model is a dynamic network flow problem with additional variables for the number and direction of used lanes and with additional complicating constraints. Because of the size of the time-expanded network, the computational effort required by standard software is already very high for tiny instances. To deal with realistic instances we propose a heuristic approach.

Bitwise Algorithm for Fastest Paths in Time-Dependent Networks

An algorithm operating on a new and compact representation of link exit times to solve the one-to-all fastest path problem in dynamic networks is presented. The new space-efficient representation that exploits the first-in, first-out property of link travel times in dynamic networks is developed. The algorithm is described, and its working on a simple network is presented. The algorithm directly operates on bit streams of data representing time-dependent link travel times and performs logical operations on 0 and 1 bits only. Computational results are presented for large dynamic networks. The ideas behind the development of this algorithm are extendable to the development of algorithms operating at the bit level to solve other variants of dynamic shortest path problems and other dynamic network flow problems. These developments then open new research possibilities in the development of solution algorithms for dynamic network problems.

Dynamic Network Flow with Uncertain Arc Capacities: Decomposition Algorithm and Computational Results

In a multiperiod dynamic network flow problem, we model uncertain arc capacities using scenario aggregation. This model is so large that it may be difficult to obtain optimal integer or even continuous solutions. We develop a Lagrangian decomposition method based on the structure recently introduced in G.D. Glockner and G.L. Nemhauser, Operations Research, vol. 48, pp. 233–242, 2000. Our algorithm produces a near-optimal primal integral solution and an optimum solution to the Lagrangian dual. The dual is initialized using marginal values from a primal heuristic. Then, primal and dual solutions are improved in alternation. The algorithm greatly reduces computation time and memory use for real-world instances derived from an air traffic control model.

A Dynamic Network Flow Problem with Uncertain arc Capacities: Formulation and Problem Structure

We consider a dynamic network flow problem where the arc capacities are random variables. This gives a multistage stochastic linear program. We describe the randomness using a multi-scenario approach. Because of the multilayered structure, there are several ways to decompose the linear program. We classify different decomposition schemes, and we develop a scheme called compath decomposition, which is derived from path decomposition for network flows. We give a polynomial-time algorithm to find a cheapest compath that can solve the subproblems resulting from compath decomposition. Finally, compath decomposition is extended to multicommodity flow problems.

The Quickest Transshipment Problem

A dynamic network consists of a graph with capacities and transit times on its edges. The quickest transshipment problem is defined by a dynamic network with several sources and sinks; each source has a specified supply and each sink has a specified demand. The problem is to send exactly the right amount of flow out of each source and into each sink in the minimum overall time. Variations of the quickest transshipment problem have been studied extensively; the special case of the problem with a single sink is commonly used to model building evacuation. Similar dynamic network flow problems have numerous other applications; in some of these, the capacities are small integers and it is important to find integral flows. There are no polynomial-time algorithms known for most of these problems. In this paper we give the first polynomial-time algorithm for the quickest transshipment problem. Our algorithm provides an integral optimum flow. Previously, the quickest transshipment problem could only be solved efficiently in the special case of a single source and single sink.

One, two, three, many, or: complexity aspects of dynamic network flows with dedicated arcs

A dynamic network consists of a directed graph with a source s, a sink t and capacities and integral transit times on the arcs. We investigate the computational complexity of dynamic network flow problems where sending flow along an arc blocks this arc as long as the transmission continues. Such arcs are called dedicated arcs. We are mainly interested in questions of the type “Given an integral time bound T and an integral flow value v, is it possible to transmit v flow units from the source s to the sink t within T time units?”. The complexity of this question strongly depends on whether the values T and v are encoded in binary, in unary, or are constant. We provide a complete classification of all variants of this problem from the computational complexity point of view. Our results establish a sharp borderline between easy and difficult cases of this dynamic network flow problem. We prove that in the dedicated arc model it is NP-hard to find the maximum flow value v that can be transmitted within T=3 time units, whereas the maximum flow value v for T=2 can be computed in polynomial time. Moreover, we prove that it is NP-hard to find the minimum time T during which v=2 units of flow can be transmitted, whereas the corresponding question for v=1 can be answered in polynomial time. Finally, we prove that the variant where T is encoded in unary and where v is not part of the input can be solved in polynomial time.

A Study of General Dynamic Network Programs with Arc Time-Delays

Dynamic network flow problems in which there is a delay on the flows in the arcs have been in existence since the early days of modern optimization. However, most previous work in this area has only considered models which are discrete in the time variable. In this paper we present a continuous-time model for a very broad class of dynamic network problems with arc time-delays. The model is a direct extension of the separated continuous linear program (SCLP) to include time-delays and is called the separated continuous linear program with time-delays (SCLPTD). By suitable transformations we are able to rewrite SCLPTD in a manner which is very close to SCLP itself. This then allows us to use all the recent theory and algorithms for SCLP to derive similar results for SCLPTD. In particular, the theory we present includes a characterization of the extreme-point solutions, an existence theorem for piecewise analytic optimal extreme-point solutions, and a strong duality theorem. We also present a class of convergent algorithms for the solution of SCLPTD in certain instances.

On the use of lexicographic min cost flows in evacuation modeling

Building evacuation problems can be represented as dynamic network-flow problems [3]. The underlying network structure of a building evolves through time yielding a time-expanded network (a dynamic network). Usually in such evacuation problems involving time, more than one objective function is appropriate. For example, minimizing the total evacuation time and evacuating a portion of the building as early as possible are two such objectives. In this article we show that lexicographical optimization is applicable in handling such multiple objectives. Minimizing the total evacuation time while avoiding cyclic movements in a building and “priority evacuation” are treated as lexicographical min cost flow problems.