An Improved Algorithm of Network Maximum Flow Based on Network Flow Matrix

The biproportional apportionment problem (BAP) must be faced in many proportional electoral systems where seats must be allocated to parties within regions. BAP is a non-trivial optimization problem, and only sophisticated algorithms are currently available for solving it. The issue is: are they “writable” as an actual law? Citizens rightly demand simple, easy to understand, voting systems. The alternative, though, seems to have simple, but unsound electoral laws. We propose the following way out of this dilemma: leave to a mathematically sophisticated algorithm the task of producing an optimal apportionment, but attach to it a “certificate of optimality”, that is, describe a simple procedure whereby anybody can check, through some elementary operations, that the seat allocation output by the algorithm is indeed an optimal apportionment. We discuss one such certificate, based on the Max flow- min cut Theorem, relative to a parametric max flow method of ours for BAP.

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.

Purpose. The purpose of the article is to develop a universal unified parallel synchronous algorithm for the implementation of tasks for calculation of maximum one- and multicommodity flows, as well as the creation of a software complex that provides the formation of surface graph models of flows and performs optimal planning of non-uniform flows in transport and other networks. Methodology. The paper investigates the possibilities of previously created and comprehensively verified heuristic parallel synchronous algorithm for calculating maximum one- and multicommodity flows in the networks, establishes its potential limitations, and determines additional advanced procedures that transform the algorithm into a universal parallel algorithm. The proposed parallel synchronous algorithm uses a width-first search strategy while simultaneously identifying possible paths of flows through the network with an estimation of their throughput. Herewith the possibility of analyzing several incremental flows across the network in one iteration was studied. Findings. The article proposes a universal unified parallel synchronous algorithm for calculating maximum flows in networks and develops a unified procedure and software package for planning of non-uniform as well as competitive flows in transport and other networks. The developed software complex implements the problems of formation of surface graph models of networks, for which the problem of optimal planning of non-uniform and competitive multicriteria flows in transport networks is solved. Originality. The article develops a new universal unified parallel synchronous algorithm and procedure for the calculation of optimal uniform, multicommodity and competitive flows in transport networks. Practical value. The practical value of the obtained results is determined by the universal capabilities and efficiency of the procedure for planning non-uniform flows in the networks based on the application of a new parallel synchronous algorithm, as well as the developed software complex, which provides the ability to solve the problems of analysis and planning of uniform and multicommodity flows in transport networks, as well as the implementation of calculation tasks of competitive models of transport and information flows formation. The software complex has a built-in editor of interactive network modeling and a toolbar, which provides both creation of new and downloading existing graphs of networks from the modeling libraries, preservation of optimum flows in the network in the form of an image and a text file, output of errors when working with the program.

The network flows approach for matrices with given row and column sums

Convex transportation problems represent an important special class of convex network flow problems, with particular applications in telecommunications, economics and engineering. This paper describes an implementation of an “exact” Newton algorithm for separable convex transportation problems. Previously in the literature, only approximate Newton algorithms had been described for the general case of convex network flow problems. The special structure of the transportation problem is used to simplify computation of dual multiplier estimates. In general, this would require factorization of a square matrix of dimension n + m − 1, were n is the number of sources and m is the number of sinks. By exploiting structure, the work of factorization is limited to a square matrix of dimension n − 1 or m − 1, whichever is smaller. Some computational experience is described. We also note an analogy between this Newton algorithm for convex problem and a particular variant of Karmarkar's algorithm for linear problems.

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.).

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.

This paper proposes a maximum-flow-based complex network approach for the analysis of the vulnerability of power systems. A new centrality index is proposed, taking into consideration the maximum flow from the source (generator) nodes to the sink (load) nodes, for assessing the network. The Max-Flow Min-Cut Theorem, also known as Ford-Fulkerson Theorem, is used for evaluating the capacity of links. The proposed methodology is then used to identify vulnerable lines of the IEEE 118 bus system and its effectiveness is demonstrated through simulation studies.

Reconstruction Approach to Reliability and Maximum Flows in Networks

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

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 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.

Polymatroidal flows with lower bounds

In this paper, we propose a novel method named Contracting Community Approach (CCA) to get the maximum flow of flow network. Firstly, we contract communities in the original network. Then, we apply classic algorithms on the contracted network to approximately solve the maximum flow problem. Experimental results show that the efficiency of the proposed algorithm. For sparse networks, the size of network is reduced to 58.38% averagely and the correctness of maximum flow is over 95%. For middle dense networks, the size of network is reduced to 65.77% averagely. For dense networks, the size of network is reduced to 64.84% averagely. And the correctness of maximum flow even reach 100% both in many middle dense and dense cases in our experiments.

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