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.     

On Solving Quickest Time Problems in Time-Dependent, Dynamic Networks

In this paper, a pseudopolynomial time algorithm is presented for solving the integral time-dependent quickest flow problem (TDQFP) and its multiple source and sink counterparts: the time-dependent evacuation and quickest transshipment problems. A more widely known, though less general version, is the quickest flow problem (QFP). The QFP has historically been defined on a dynamic network, where time is divided into discrete units, flow moves through the network over time, travel times determine how long each unit of flow spends traversing an arc, and capacities restrict the rate of flow on an arc. The goal of the QFP is to determine the paths along which to send a given supply from a single source to a single sink such that the last unit of flow arrives at the sink in the minimum time. The main contribution of this paper is the time-dependent quickest flow (TDQFP) algorithm which solves the TDQFP, i.e. it solves the integral QFP, as defined above, on a time-dependent dynamic network, where the arc travel times, arc and node capacities, and supply at the source vary with time. Furthermore, this algorithm solves the time-dependent minimum time dynamic flow problem, whose objective is to determine the paths that lead to the minimum total time spent completing all shipments from source to sink. An optimal solution to the latter problem is guaranteed to be optimal for the TDQFP. By adding a small number of nodes and arcs to the existing network, we show how the algorithm can be used to solve both the time-dependent evacuation and the time-dependent quickest transshipment problems. This revised version was published online in August 2006 with corrections to the Cover Date.     

Maximum-throughput dynamic network flows

This paper presents and solves the maximum throughput dynamic network flow problem, an infinite horizon integer programming problem which involves network flows evolving over time. The model is a finite network in which the flow on each arc not only has an associated upper and lower bound but also an associated transit time. Flow is to be sent through the network in each period so as to satisfy the upper and lower bounds and conservation of flow at each node from some fixed period on. The objective is to maximize the throughput, the net flow circulating in the network in a given period, and this throughput is shown to be the same in each period. We demonstrate that among those flows with maximum throughput there is a flow which repeats every period. Moreover, a duality result shows the maximum throughput equals the minimum capacity of an appropriately defined cut. A special case of the maximum dynamic network flow problem is the problem of minimizing the number of vehicles to meet a fixed periodic schedule. Moreover, the elegantsolution derived by Ford and Fulkerson for the finite horizon maximum dynamic flow problem may be viewed as a special case of the infinite horizon maximum dynamic flow problem and the optimality of solutions which repeat every period.     

Flows over time in time-varying networks: Optimality conditions and strong duality

There has been much research on network flows over time due to their important role in real world applications. This has led to many results, but the more challenging continuous time model still lacks some of the key concepts and techniques that are the cornerstones of static network flows. The aim of this paper is to advance the state of the art for dynamic network flows by developing the continuous time analogues of the theory for static network flows. Specifically, we make use of ideas from the static case to establish a reduced cost optimality condition, a negative cycle optimality condition, and a strong duality result for a very general class of network flows over time.     

A new pivot selection rule for the network simplex algorithm

We present a new network simplex pivot selection rule, which we call theminimum ratio pivot rule, and analyze the worst-case complexity of the resulting network simplex algorithm. We consider networks withn nodes,m arcs, integral arc capacities and integral supplies/demands of nodes. We define a {0, 1}-valued penalty for each arc of the network. The minimum ratio pivot rule is to select that eligible arc as the entering arc whose addition to the basis creates a cycle with the minimum cost-to-penalty ratio. We show that the so-defined primal network simplex algorithm solves minimum cost flow problem within O(nΔ) pivots and in O(Δ(m + n logn)) time, whereΔ is any upper bound on the sum of all arc flows in every feasible flow. For assignment and shortest path problems, our algorithm runs in O(n ²) pivots and O(nm +n ² logn) time.     

A Generalization of a Theorem of Liao

Let X be a metrizable space and let φ：R× X → X be a continuous flow on X. For any given {φt}-invariant Borel probability measure, this paper presents a {φt}-invariant Borel subset of X satisfying the requirements of the classical ergodic theorem for the contiImous flow （X, {φt}）. The set is more restrictive than the ones in the literature, but it might be more useful and convenient, particularly for non-uniformly hyperbolic systems and skew-product flows.     

A Theory of Complexity for Continuous Time Systems

We present a model of computation with ordinary differential equations (ODEs) which converge to attractors that are interpreted as the output of a computation. We introduce a measure of complexity for exponentially convergent ODEs, enabling an algorithmic analysis of continuous time flows and their comparison with discrete algorithms. We define polynomial and logarithmic continuous time complexity classes and show that an ODE which solves the maximum network flow problem has polynomial time complexity. We also analyze a simple flow that solves the Maximum problem in logarithmic time. We conjecture that a subclass of the continuous P is equivalent to the classical P.     

Approximating earliest arrival flows with flow-dependent transit times

For the earliest arrival flow problem one is given a network G=(V,A) with capacities u(a) and transit times τ(a) on its arcs a∈A, together with a source and a sink vertex s,t∈V. The objective is to send flow from s to t that moves through the network over time, such that for each time θ∈[0,T) the maximum possible amount of flow up to this time reaches t. If, for each θ∈[0,T), this flow is a maximum flow for time horizon θ, then it is called earliest arrival flow. In practical applications a higher congestion of an arc in the network often implies a considerable increase in transit time. Therefore, in this paper we study the earliest arrival problem for the case that the transit time of each arc in the network at each time θ depends on the flow on this particular arc at that time θ.For constant transit times it has been shown by Gale that earliest arrival flows exist for any network. We give examples, showing that this is no longer true for flow-dependent transit times. For that reason we define a relaxed version of this problem where the objective is to find flows that are almost earliest arrival flows. In particular, we are interested in flows that, for each θ∈[0,T), need only α-times longer to send the maximum flow to the sink. We give both constant lower and upper bounds on α; furthermore, we present a constant factor approximation algorithm for this problem.     

On the enrouting protocol problem under uncertainty

In this paper we present a two-stage stochastic mixed 0–1 dynamic multicommodity model and algorithm for determining the enrouting protocol in the telecommunications network under uncertainty. Given the network connectivity, node processing and buffer and arc flow capacity, the aim is to determine the outgoing arc for the information flow reaching a given node for each destination terminal node (i.e., obtaining the route to be followed by the information flow from each origin terminal node to each destination terminal node). The origin–destination (O–D) flow matrix is given by the number of information packets to be sent from the origin terminal nodes to the destination terminal nodes along a given time horizon, i.e., a call scale. The uncertainty in the O–D flow matrix is treated via a scenario tree approach. The main goal is to minimize a composite function of the expected lost information, a penalization of the deviation from the FIFO strategy on the information flow entering the network, and the expected number of nodes visited by the information packets. A mixture of an enrouting arc generation scheme and a genetic algorithm for obtaining the enrouting protocols over the scenarios is presented. The tool presented in this paper could be used for simulating the enrouting protocols to analyze the saturation of the network, but it has a time constraint for real time operation. Faster algorithms are needed to define the routing tables during the operation stage. Computational experience is reported.     

Chaos in a dynamic model of urban transportation network flow based on user equilibrium states

In this study, we investigate the dynamical behavior of network traffic flow. We first build a two-stage mathematical model to analyze the complex behavior of network flow, a dynamical model, which is based on the dynamical gravity model proposed by Dendrinos and Sonis [Dendrinos DS, Sonis M. Chaos and social-spatial dynamic. Berlin: Springer-Verlag; 1990] is used to estimate the number of trips. Considering the fact that the Origin–Destination (O–D) trip cost in the traffic network is hard to express as a functional form, in the second stage, the user equilibrium network assignment model was used to estimate the trip cost, which is the minimum cost of used path when user equilibrium (UE) conditions are satisfied. It is important to use UE to estimate the O–D cost, since a connection is built among link flow, path flow, and O–D flow. The dynamical model describes the variations of O–D flows over discrete time periods, such as each day and each week. It is shown that even in a system with dimensions equal to two, chaos phenomenon still exists. A “Chaos Propagation” phenomenon is found in the given model.     

A demand model with departure time choice for within-day dynamic traffic assignment

A within-day dynamic demand model is formulated, embodying, in addition to the classic generation, distribution and modal split stages, an actual demand model taking into account departure time choice. The work focuses on this last stage, represented through an extension of the discrete choice framework to a continuous choice set. The dynamic multimodal supply and equilibrium model based on implicit path enumeration, which have been developed in previous work are outlined here, to define within-day dynamic elastic demand stochastic multimodal equilibrium as a fixed point problem on users flows and transit line frequencies. A MSA algorithm capable, in the case of Logit route choice models, of supplying equilibrium flows and frequencies on real dimension networks, is presented, as well as the specific procedures implementing the departure time choice and actual demand models. Finally, the results obtained on a test network are presented and conclusions are drawn.     

Minimum Maximal Flow Problem: An Optimization over the Efficient Set

The network flow theory and algorithms have been developed on the assumption that each arc flow is controllable and we freely raise and reduce it. We however consider in this paper the situation where we are not able or allowed to reduce the given arc flow. Then we may end up with a maximal flow depending on the initial flow as well as the way of augmentation. Therefore the minimum of the flow values that are attained by maximal flows will play an important role to see how inefficiently the network can be utilized. We formulate this problem as an optimization over the efficient set of a multicriteria program, propose an algorithm, prove its finite convergence, and report on some computational experiments.     

Properties of measures on totally ordered spaces

The aim of this paper is to study the properties of measures on a totally ordered spaceX. It is shown that if the cardinal of every closed discrete subset ofX is of measure zero then every non-atomic measure onX is τ-additive. It is also proved that a continuous Borel measure μ onX is Radon if and only if μ is perfect, τ-additive and the support of μ is almost a totally ordered space.

     

Measurable group actions are essentially Borel actions

If a locally compact groupG acts on a Lebesgue probability space (X, λ), it is natural to consider these conditions: (a) each group element preserves the class of λ, and (b) the action function is measurable. The latter is a weakening of the requirement that the action be Borel, providedX has a particular Borel structure as well as the σ-algebra of measurable sets. In this paper, we give an example showing that such an action need not be Borel relative to the given Borel structure, and prove that there is always a conull invariant subset and a new standard Borel structure on that subset for which the action is Borel. This is the meaning of the title. Supported by a University of Colorado Faculty Fellowship.     

A logistics problem: a non-markovian queueing network model and algorithm

In this paper, a stochastic version of the classical deterministic balanced single commodity capacitated transportation network problem is presented. In this model, each arc of the network connects a supply node to a demand node and the flow of units forming along each arc of the network forms a stochastic process (i.e.G/M/1 queueing system with generally distributed interarrival time, a Markovian server, a single server, infinite capacity, and the first come first served queueing discipline). In this model, the total transportation cost is minimized such that the total supply rate is equal to the total demand rate, and the resulting probability of finding excessive congestion along each arc (i.e., the resulting probability of finding congestion inside the queueing system formed along each arc in excess of a fixed number) is equal to a desirable value     

Über mittlere Rückkehrzeit unter einer ma\treuen Strömung

Summary Let T be a one-parameter semigroup of measure preserving transformations of a probability space. The theorem of Kac on mean recurrence time of the points x in a measurable subset E under a discrete semigroup is carried over to the case of a flow T with continuous time parameter t0. Recurrence time is defined as the infimum of these parameter values t>0 for which the orbit of x has returned to E after having temporarily left the set E. The results are first formulated for a probability space without any topological structure; they are then applied to the case of a continuous flow in a compact metric space.     

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.     

Optimum synthesis of discrete capacitated networks with multi-terminal commodity flow requirements

Network design problems arise in a wide range of applied areas including telecommunications, computer networks, and transportation. In this paper, we address the following discrete capacitated multi-terminal network design problem. Given a connected digraph G = (V,A), a set of L potential facilities to be installed on each arc, and a set of K multi-terminal (non-simultaneous) commodity flow requirements, the problem is to find a set of facilities to install in order to route the K nonsimultaneous flows while minimizing the total fixed plus variable costs. We describe an exact procedure for solving this problem based on Benders decomposition. Our algorithm includes several features that significantly improve the efficiency of the basic approach. Computational results attest to the efficacy of the proposed algorithm, which can solve medium- to large-scale problems to optimality.     

The flow circulation sharing problem

The flow circulation sharing problem is defined as a network flow circulation problem with a maximin objective function. The arcs in the network are partitioned into regular arcs and tradeoff arcs where each tradeoff arc has a non-decreasing tradeoff function associated with it. All arcs have lower and upper bounds on their flow while the value of the smallest tradeoff function is maximized. The model is useful in equitable resource allocation problems over time which is illustrated in a coal strike example and a submarine assignment example. Some properties including optimality conditions are developed. Each cut in the network defines a knapsack sharing problem which leads to an optimality condition similar to the max flow/min cut theorem. An efficient algorithm for both the continuous and integer versions of the flow circulation sharing problem is developed and computational experience given. In addition, efficient algorithms are developed for problems where some of the arcs have infinite flow upper bounds.     

Optimizing cost flows by edge cost and capacity upgrade

We examine a network upgrade problem for cost flows. A budget can be distributed among the arcs of the network. An investment on each single arc can be used either to decrease the arc flow cost, or to increase the arc capacity, or both. The goal is to maximize the flow through the network while not exceeding bounds on the budget and on the total flow cost.

The problems are NP-hard even on series-parallel graphs. We provide an approximation algorithm on series-parallel graphs which, for arbitrary δ,>0, produces a solution which exceeds the bounds on the budget and the flow cost by factors of at most 1+δ and 1+, respectively, while the amount of flow is at least that of an optimum solution. The running time of the algorithm is polynomial in the input size and 1/(δ). In addition we give an approximation algorithm on general graphs applicable to problem instances with small arc capacities.