A new Karzanov-type O( n3) max-flow algorithm

In 1961, Gomory and Hu showed that the All-Pairs Max-Flow problem of computing the max-flow between all $\begin{pmatrix}n\\2\end{pmatrix}$ pairs of vertices in an undirected graph can be solved using only $n-1$ calls to any (single-pair) max-flow algorithm. Even assuming a linear-time max-flow algorithm, this yields a running time of $O(mn)$, which is $O(n^{3})$ when $m=\Theta(n^{2})$. While subsequent work has improved this bound for various special graph classes, no subcubic-time algorithm has been obtained in the last 60 years for general graphs. We break this longstanding barrier by giving an $\tilde{O}(n^{2})$-time algorithm on general, integer-weighted graphs. Combined with a popular complexity assumption, we establish a counter-intuitive separation: all-pairs max-flows are strictly easier to compute than all-pairs shortest-paths.Our algorithm produces a cut-equivalent tree, known as the Gomory-Hu tree, from which the max-flow value for any pair can be retrieved in near-constant time. For unweighted graphs, we refine our techniques further to produce a Gomory-Hu tree in the time of a poly-logarithmic number of calls to any maxflow algorithm. This shows an equivalence between the all-pairs and single-pair max-flow problems, and is optimal up to polylogarithmic factors. Using the recently announced $m^{1+o(1)}$-time max-flow algorithm (Chen et al., March 2022), our Gomory-Hu tree algorithm for unweighted graphs also runs in $m^{1+o(1)}$-time.

Supply chain management has recently renovated its strategy by implementing a cross-docking scheme. Cross-docking is a calculated logistics strategy where freight emptied from inbound vehicles is handled straightforwardly onto outbound vehicles, eliminating the intermediate storage process. The cross-docking approach thrives on the minimum storage time of goods in the inventory. Most of the cross-docks avail temporary storage docks where items can be stored for up to 24 hours before being packed up for transportation. The storage capacity of the cross-dock varies depending on the nature of ownership. In the rented cross-docks center, the temporary storage docks are considered of infinite capacity. This study believes that the temporary storage facilities owned by the cross-dock center are of finite capacity, which subsequently affects the waiting time of the goods. The flow rate of goods within the cross-docks is expected to be maximum to avoid long waiting for goods in the queue. This paper uses a series of max-flow algorithms, namely Ford Fulkerson, Edmond Karp, and Dinic's, to optimize the flow of goods between the inbound port and the outbound dock and present a logical explanation to reduce the waiting time of the trucks. A numerical example is analyzed to prove the efficacity of the algorithm in finding maximum flow. The result demonstrates that Dinic's algorithm performs better than the Ford Fulkerson and Edmond Karp algorithm at addressing the problem of maximum flow at the cross-dock. The algorithm effectively provided the best result regarding iteration and time complexity. In addition, it also suggested the bottleneck paths of the network in determining the maximum flow.

Parametric max flow problems in a class of networks with series-parallel structure

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.).

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.

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

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

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.

For guaranteeing all multicast destination nodes receiving the source information with their maximal flow respectively and obtaining the network maximal throughput, a heuristic algorithm based on network coding, Maximal Flow for Single-source Two-destinations Multicast (MFSTM) is proposed to maximize the network throughput. By calculating the each destination’s maximal flow, the number of link-disjoint paths which equals to destination’s maximal flow, are searched for each destination to construct the network coding graph. A heuristic algorithm based on network coding is designed to delete the redundant link in the network coding graph and guarantee the network throughput maximization. Comparing the traditional maximal multicast stream algorithm based on network coding, the simulation results show that the MFSTM algorithm makes two destinations receive the information at the speed of their maximal flow respectively, and decode the source node information at each destination node successfully.

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

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.

We investigate how to rapidly solve an online sequence of maximum flow problems (MFPs). Such sequences arise in a diverse collection of settings including stochastic network programming and constraint programming. In this paper, we formalize the study of solving a sequence of MFPs, introduce a maximum flow algorithm designed for "warm starts" and extend our work to computing a robust minimum cut. We demonstrate that our algorithms reduce the running time by an order of magnitude when compared similar codes that use a black-box MFP solver. In particular, we show that our algorithm for robust minimum cuts can solve instances in seconds that would require over four hours using a black-box maximum flow solver.

The objective of the study is to present an overview of network analysis with a special focus on the construction and application of complete graphs in organizational networks. Network analysis is approached both statistically and algorithmically to understand the relationships among system components. In this context, complete graphs allow for the effective implementation of various algorithms to solve the critical path, minimum spanning tree (MST), maximal flow and shortest route problems. These problems are addressed using different classical graph theory techniques such as the PERT (Program Evaluation and Review Technique) method to solve the critical path, Prim’s algorithm for finding the MST, the Ford-Fulkerson algorithm for computing the maximum flow in a network, and the traveling salesman algorithm for determining the shortest route of a complete graph with n vertices in a single network design with an appropriate numerical example. The comprehensive connectivity in complete graphs ensures greater flexibility and accuracy in solving complex network-related problems in organizational and operational settings.

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