An introduction to dynamic generative networks: Minimum cost flow

What we are dealing with is a class of networks called dynamic generative network flows in which the flow commodity is dynamically generated at source nodes and dynamically consumed at sink nodes. As a basic assumption, the source nodes produce the flow according to time generative functions and the sink nodes absorb the flow according to time consumption functions. This paper tries to introduce these networks and formulate minimum cost dynamic flow problem for a pre-specified time horizon T. Finally, some simple, efficient approaches are developed to solve the dynamic problem, in the general form when the capacities and costs are time varying and some other special cases, as a minimum cost static flow problem.     

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.     

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.     

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.     

A multi-population hybrid biased random key genetic algorithm for hop-constrained trees in nonlinear cost flow networks

Genetic algorithms and other evolutionary algorithms have been successfully applied to solve constrained minimum spanning tree problems in a variety of communication network design problems. In this paper, we enlarge the application of these types of algorithms by presenting a multi-population hybrid genetic algorithm to another communication design problem. This new problem is modeled through a hop-constrained minimum spanning tree also exhibiting the characteristic of flows. All nodes, except for the root node, have a nonnegative flow requirement. In addition to the fixed charge costs, nonlinear flow dependent costs are also considered. This problem is an extension of the well know NP-hard hop-constrained Minimum Spanning Tree problem and we have termed it hop-constrained minimum cost flow spanning tree problem. The efficiency and effectiveness of the proposed method can be seen from the computational results reported.     

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.     

Minimum flow problem on network flows with time-varying bounds

In this paper, we consider the minimum flow problem on network flows in which the lower arc capacities vary with time. We will show that this problem for set {0, 1, … , T} of time points can be solved by at most n minimum flow computations, by combining of preflow-pull algorithm and reoptimization techniques (no matter how many values of T are given). Running time of the presented algorithm is O(n²m).     

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.     

Stability with file arrivals and departures in multichannel cellular wireless networks

This paper studies scheduling in multichannel wireless networks with flow-level dynamics. We consider a downlink network with a single base station, M channels (frequency bands), and multiple mobile users (flows). We also assume mobiles dynamically join the network to receive finite-size files and leave after downloading the complete files. A recent study van de Ven et al. (in Proc. IEEE Infocom., pp. 1701?C1709, 2009) has shown that the MaxWeight algorithm fails to be throughput-optimal under these flow-level dynamics. The main contribution of this paper is the development of joint channel-assignment and workload-based scheduling algorithms for multichannel downlink networks with dynamic flow arrivals/departures. We prove that these algorithms are throughput-optimal. Our simulations further demonstrate that a hybrid channel-assignment and workload-based scheduling algorithm significantly improves the network performance (in terms of both file-transfer delay and blocking probability) compared to the existing algorithms.     

A Fast Algorithm For Finding A Maximum Free Multiflow In An Inner Eulerian Network And Some Generalizations

be a network, where is an undirected graph with nodes and edges, is a set of specified nodes of , called terminals, and each edge of has a nonnegative integer capacity . If the total capacity of edges with one end at is even for every non-terminal node , then is called inner Eulerian. A free multiflow is a collection of flows between arbitrary pairs of terminals such that the total flow through each edge does not exceed its capacity. In this paper we first generalize a method in Karzanov [11] to find a maximum integer free multiflow in an inner Eulerian network, in time, where is the complexity of finding a maximum flow between two terminals. Next we extend our algorithm to solve the so-called laminar locking problem on multiflows, also in time. We then consider analogs of the above problems in inner balanced directed networks, which means that for each non-terminal node , the sums of capacities of arcs entering and leaving are the same. We show that for such a network a maximum integer free multiflow can be constructed in time, and then extend this result to the corresponding locking problem. Received: March 24, 1997     

Second best temporally repeated flows

FORD and FULKERSON have shown that a stationary maximal dynamic flow can be obtained by solving a transhipment problem associated with the static network and thereby finding the maximal temporally repeated dynamic flow. This flow is known to be an optimal dynamic flow. This paper presents an algorithm for second best temporally repeated flows. A numerical example is presented.     

Estimation of flows in flow networks

Let G be a directed graph with an unknown flow on each edge such that the following flow conservation constraint is maintained: except for sources and sinks, the sum of flows into a node equals the sum of flows going out of a node. Given a noisy measurement of the flow on each edge, the problem we address, which we call the Most Probable Flow Estimation problem (MPFE), is to estimate the most probable assignment of flow for every edge such that the flow conservation constraint is maintained. We provide an algorithm called ΔY-mpfe for solving the MPFE problem when the measurement error is Gaussian (Gaussian-MPFE). The algorithm works in O(∣E∣ + ∣V∣²) when the underlying undirected graph of G is a 2-connected planar graph, and in O(∣E∣ + ∣V∣) when it is a 2-connected serial-parallel graph or a tree. This result is applicable to any Minimum Cost Flow problem for which the cost function is τ_e(X_e − μ_e)² for edge e where μ_e and τ_e are constants, and X_e is the flow on edge e. We show that for all topologies, the Gaussian-MPFE’s precision for each edge is analogous to the equivalent resistance measured in series to this edge in an electrical network built by replacing every edge with a resistor reflecting the measurement’s precision on that edge.     

An algorithm for minimal dynamic flow

FORD and FULKERSON have shown that a stationary maximal dynamic flow can be obtained by solving a transhipment problem associated with the static network and thereby finding the maximal temporally repeated dynamic flow. This flow is known to be an optimal dynamic flow. This paper presents the remark that temporally repeated flows may be not optimal for a minimal dynamic flow and an algorithm for such a flow. A numerical example is presented.     

Continuous and discrete flows over time

Network flows over time form a fascinating area of research. They model the temporal dynamics of network flow problems occurring in a wide variety of applications. Research in this area has been pursued in two different and mainly independent directions with respect to time modeling: discrete and continuous time models. In this paper we deploy measure theory in order to introduce a general model of network flows over time combining both discrete and continuous aspects into a single model. Here, the flow on each arc is modeled as a Borel measure on the real line (time axis) which assigns to each suitable subset a real value, interpreted as the amount of flow entering the arc over the subset. We focus on the maximum flow problem formulated in a network where capacities on arcs are also given as Borel measures and storage might be allowed at the nodes of the network. We generalize the concept of cuts to the case of these Borel Flows and extend the famous MaxFlow-MinCut Theorem.     

The multi-terminal maximum-flow network-interdiction problem

This paper defines and studies the multi-terminal maximum-flow network-interdiction problem (MTNIP) in which a network user attempts to maximize flow in a network among K ? 3 pre-specified node groups while an interdictor uses limited resources to interdict network arcs to minimize this maximum flow. The paper proposes an exact (MTNIP-E) and an approximating model (MPNIM) to solve this NP-hard problem and presents computational results to compare the models. MTNIP-E is obtained by first formulating MTNIP as bi-level min-max program and then converting it into a mixed integer program where the flow is explicitly minimized. MPNIM is binary-integer program that does not minimize the flow directly. It partitions the node set into disjoint subsets such that each node group is in a different subset and minimizes the sum of the arc capacities crossing between different subsets. Computational results show that MPNIM can solve all instances in a few seconds while MTNIP-E cannot solve about one third of the problems in 24 hour. The optimal objective function values of both models are equal to each other for some problems while they differ from each other as much as 46.2% in the worst case. However, when the post-interdiction flow capacity incurred by the solution of MPNIM is computed and compared to the objective value of MTNIP-E, the largest difference is only 7.90% implying that MPNIM may be a very good approximation to MTNIP-E.     

The first <Emphasis Type="Italic">K</Emphasis> minimum cost paths in a time-schedule network

The time-constrained shortest path problem is an important generalisation of the classical shortest path problem and in recent years has attracted much research interest. We consider a time-schedule network, where every node in the network has a list of pre-specified departure times and departure from a node may take place only at one of these departure times. The objective of this paper is to find the first K minimum cost simple paths subject to a total time constraint. An efficient polynomial time algorithm is developed. It is also demonstrated that the algorithm can be modified for finding the first K paths for all possible values of total time.     

节点有自环的网络流数学模型

以网络流中节点有自环的情形为对象进行研究,把节点分开为入点和出点,节点自环转换为流量相同,方向相反的两条弧,改进了网络流数学模型.改进后的数学模型在处理原来网络流中节点发生异常情况时,即节点不遵守流量守恒条件时,其节点自环的流能够起特殊的调节作用.在描述网络流异常状态时,给出了网络流状态周期的阶跃性质.通过网络模型对应的邻接矩阵对网络流进行计算和监控,给出一个节点环流的应用实例.     

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.     

Sensitivity analysis of permeability parameters for flows on Barcelona networks

We consider the problem of optimizing vehicular traffic flows on an urban network of Barcelona type, i.e. square network with streets of not equal length. In particular, we describe the effects of variation of permeability parameters, that indicate the amount of flow allowed to enter a junction from incoming roads.On each road, a model suggested by Helbing et al. (2007) [11] is considered: free and congested regimes are distinguished, characterized by an arrival flow and a departure flow, the latter depending on a permeability parameter. Moreover we provide a rigorous derivation of the model from fluid dynamic ones, using recent results of Bretti et al. (2006) [3]. For solving the dynamics at nodes of the network, a Riemann solver maximizing the through flux is used, see Coclite et al. (2005) [4] and Helbing et al. (2007) [11].The network dynamics gives rise to complicate equations, where the evolution of fluxes at a single node may involve time-delayed terms from all other nodes. Thus we propose an alternative hybrid approach, introducing additional logic variables. Finally we compute the effects of variations on permeability parameters over the hybrid dynamics and test the obtained results via simulations.