A new maximal flow algorithm for solving optimization problems with linguistic capacities and flows

Augmenting path algorithms, first introduced by Ford and Fulkerson (Ford, L. R., D. R. Fulkerson. 1962. Flows in Networks. Princeton University Press, Princeton, N.J.), are widely used in optimization. Examples include Schönsleben's polymatroid intersection algorithm (Schönsleben, P. 1980. Ganzzahlige polymatroid-intersection-algorithmen. Ph.D. thesis, Eidgenössischen Technischen Hochschule, Zürich.), the maximum polymatroidal network flow algorithm of Lawler and Martel (Lawler, E. L., C. U. Martel. 1982. Computing maximal polymatroidal network flows. Math. Oper. Res. 7 334–347.), Frank's algorithm for the Edmonds–Giles polyhedron (Frank, A. 1984. Finding feasible vectors of Edmonds-Giles polyhedra. J. Combin. Theory Ser. B 36 221-239.) and Cunningham's algorithm for testing membership in matroid polyhedra (Cunningham, H. W. 1984. Testing membership in matroid polyhedra. J. Combin. Theory Ser. B 36 161–188.). Here we give an order of magnitude improvement for the above algorithms by using an approach analogous to that of Dinits’ maximum flow algorithm (Dinits, E. A. 1970. Algorithm for solution of a problem of maximum flow in a network with power estimation. Soviet Math. Dokl. 11 1277–1280.).

With an increasing number of processing elements (PEs) integrated on a single chip, fault-tolerant techniques are critical to ensure the reliability of such complex systems. In current reconfigurable architectures, redundant PEs are utilized for fault tolerance. In the presence of faulty PEs, the physical topologies of various chips may be different, so the concept of virtual topology from network embedding problem has been used to alleviate the burden for the operating systems. With limited hardware resources, how to reconfigure a system into the most effective virtual topology such that the maximum repair rate can be reached presents a significant challenge. In this article, a new approach using a maximum flow (MF) algorithm is proposed for an efficient topology reconfiguration in reconfigurable architectures. In this approach, topology reconfiguration is converted into a network flow problem by constructing a directed graph; the solution is then found by using the MF algorithm. This approach optimizes the use of spare PEs with minimal impacts on area, throughput, and delay, and thus it significantly improves the repair rate of faulty PEs. In addition, it achieves a polynomial reconfiguration time. Experimental results show that compared to previous methods, the MF approach increases the probability to repair faulty PEs by up to 50% using the same redundant resources. Compared to a fault-free system, the throughput only decreases by less than 2.5% and latency increases by less than 4%. To consider various types of PEs in a practical application, a cost factor is introduced into the MF algorithm. An enhanced approach using a minimum-cost MF algorithm is further shown to be efficient in the fault-tolerant reconfiguration of heterogeneous reconfigurable architectures.

We study the class of preflow push algorithms recently introduced by Goldberg and Tarjan for solving the maximum network flow problem on a weighted digraph G(V,E). We improve Goldberg and Tarjanis O(n3) time bound for the maximum distance preflow push algorithm to O(n2√m) and show that this bound is tight by constructing a parametrized worst case network. We then develop the maximal excess preflow push algorithm and show that it achieves a bound of O(n2√m) pushes. Based on this we develop a maximum network flow algorithm for the synchronous distributed model of computation that uses at most O(n2√m) messages and O(n2) time, thereby improving upon the best previously known algorithms for this model.

In this paper, we focus on the dynamic traffic assignment (DTA) problem on large-scale expressway networks especially under the condition of traffic events (such as severe weather, large traffic accidents etc.). We formulate the expressway network DTA problem as a nonconvex optimization problem. Considering the difficulty of solving the problem, we put forward an approximate solution algorithm based on the network flow theory with high computational efficiency and strong applicability. Specifically, a machine learning approach and a series of customization strategies of network flow model (including multi-origin and multidestination split, flow discretization and parallel edges addition methods) are proposed to transform the DTA problem into a resoluble network flow problem. After that, in order to improve the operational efficiency of an expressway network after traffic events occurring, the proposed network flow based two-layer DTA model (NTDM) is separated into two parts-traffic flow limitation model based on maximum flow algorithm and traffic assignment model based on minimum cost flow algorithm. Experiments on real expressway network data in some provinces of China show that our method can make a significant 26.03% reduction in saturation rate in a particular scenario. In efficiency test, the NTDM is at least 600 times faster than the traditional analytic method in a network with thousands of nodes.

We study the class of preflow push algorithms recently introduced by Goldberg and Tarjan for solving the maximum network flow problem on a weighted digraph G(V,E). We improve Goldberg and Tarjanis O(n3) time bound for the maximum distance preflow push algorithm to O(n2√m) and show that this bound is tight by constructing a parametrized worst case network. We then develop the maximal excess preflow push algorithm and show that it achieves a bound of O(n2√m) pushes. Based on this we develop a maximum network flow algorithm for the synchronous distributed model of computation that uses at most O(n2√m) messages and O(n2) time, thereby improving upon the best previously known algorithms for this model.

The bulk power system is one of the largest man-made networks and its size makes control an extremely difficult task. This paper presents a method to control a power network using UPFCs set to levels determined by a maximum flow (max-flow) algorithm. The graph-theory-based max-flow is applied to the power system for UPFC placement and scheduling. A distributed version of max-flow is described to coordinate the actions of the UPFCs distributed in a power network. Two sample power systems were tested using max-flow for UPFC placement and settings. The resulting system characteristics are examined over all single-line contingencies and the appropriateness of the maximum flow algorithm for power flow control is discussed.

Chapter 5 - Efficient CUDA Algorithms for the Maximum Network Flow Problem

Maximal flows reach at least a 1/2 approximation of the maximum flow in client-server networks. By adding only 1 additional time round to any distributed maximal flow algorithm we show how this 1/2-approximation can be improved on bounded-degree networks. We call these modified maximal flows ‘locally fair’ since there is a measure of fairness prescribed to each client and server in the network. Let N = (U,V,E,b) represent a client-server network with clients U, servers V, network links E, and node capacities b, where we assume that each capacity is at least one unit. Let d(u) denote the b-weighted degree of any node u ∈ U ∪ V, Δ = max {d(u) | u ∈ U } and δ = min { d(v) | v ∈ V }. We show that a locally-fair maximal flow f achieves an approximation to the maximum flow of \(\min \{ 1, \frac{\Delta^2 - \delta} {2\Delta^2 - \delta\Delta - \Delta}\) }, and this result is sharp for any given integers δ and Δ. This results are of practical importance since local-fairness loosely models the steady-state behavior of TCP/IP and these types of degree-bounds often occur naturally (or are easy to enforce) in real client-server systems.KeywordsDistributed Flow AlgorithmsOblivious RoutingNetwork AlgorithmsMaximal Flow

In this paper, A comparative study is made on the determination of maximum flow in a network by Maximal Flow Algorithm and Bounded Variable Simplex Method. We have introduced a technique to find the route of a flow in any iteration of the bounded variable simplex algorithm. Dhaka Univ. J. Sci. 61(1): 19-26, 2013 (January) DOI: http://dx.doi.org/10.3329/dujs.v61i1.15091

The capacity (or maximum flow) of an unicast network is known to be equal to the minimum s-t cut capacity due to the max-flow min-cut theorem. If the topology of a network (or link capacities) is dynamically changing or unknown, it is not so trivial to predict statistical properties on the maximum flow of the network. In this paper, we present a probabilistic analysis for evaluating the accumulate distribution of the minimum s-t cut capacity on random graphs. The graph ensemble treated in this paper consists of weighted graphs with arbitrary specified degree distribution. The main contribution of our work is a lower bound for the accumulate distribution of the minimum s-t cut capacity. From some computer experiments, it is observed that the lower bound derived here reflects the actual statistical behavior of the minimum s-t cut capacity of random graphs with specified degrees.

The maximum flow problem is an important problem of network optimization and it covers a wide range of engineering and management applications. The goal of the problem is to find the maximum amount of flow from the source to the sink in a network. This paper investigates two models of the maximum flow of an uncertain random network under the framework of chance theory. The expected value constrained maximum flow and chance constrained maximum flow models with uncertain random arc capacities are proposed. We prove that there exists an equivalence relationship between the uncertain random maximum flow models and the deterministic maximum flow models. Furthermore, some important properties of the models are analyzed, and two algorithms are proposed. Finally, as an illustration, a numerical example is presented to show the effectiveness of the models and algorithms.

We present an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with m edges and polynomially bounded integral demands, costs, and capacities in m 1+ o (1) time. Our algorithm builds the flow through a sequence of m 1+ o (1) approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized m o (1) time using a new dynamic graph data structure. Our framework extends to algorithms running in m 1+ o (1) time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives almost-linear time algorithms for several problems including entropy-regularized optimal transport, matrix scaling, p -norm flows, and p -norm isotonic regression on arbitrary directed acyclic graphs.

This paper investigates the application of a minimum loss path finding algorithm to determine the maximum flow in generalized networks that are characterized by arc losses or gains. In these generalized network flow problems, each arc has not only a defined capacity but also a loss or gain factor, which must be taken into consideration when calculating the maximum achievable flow. This extension of the traditional maximum flow problem requires a more comprehensive approach, where the maximum amount of flow is determined by accounting for additional factors such as costs, varying arc capacities, and the specific loss or gain associated with each arc. This paper extends the classic Ford–Fulkerson algorithm, adapting it to iteratively identify source-to-sink (s − t) residual directed paths with minimum cumulative loss and generalized augmenting paths (GAPs), thus enabling the efficient computation of maximum flow in such complex networks. Moreover, to enhance the computational performance of the proposed algorithm, we conducted extensive studies on parallelization techniques using graphics processing units (GPUs). Significant improvements in the algorithm’s efficiency and scalability were achieved. The results demonstrate the potential of GPU-accelerated computations in handling real-world applications where generalized network flows with arc losses and gains are prevalent, such as in telecommunications, transportation, or logistics networks.

<p>In this paper, we solve this problem of finding maximum flow in extended mixed network by Revised preflow-push methods of Goldberg This algorithm completely different algorithm postflow-pull in [15]. However, we share some common theory with [15].</p>

The maximum flow problem is a fundamental problem in graph theory and combinatorial optimization with a variety of important applications. Known distributed algorithms for this problem do not tolerate faults or adjust to dynamic changes in network topology. This paper presents the first distributed self-stabilizing algorithm for the maximum flow problem. Starting from an arbitrary state, the algorithm computes the maximum flow in a acyclic network in finitely many steps. Since the algorithm is self-stabilizing, it is inherently tolerant to transient faults and can automatically adjust to topology changes and to changes in other parameters of the problem. A slight modification of the original algorithm is also presented and it is conjectured that the new algorithm computes a maximum flow in arbitrary networks. >