On the planar integer two-flow problem

A definitive classification is established for all conceivable and permissible duality states of two slightly different versions (Strang and Iri) of infinite dimensional max–flow min–cut dual optimization problems. The approach builds on three elements: (1) previous work of the first author on duality gaps in max–flow problems under appropriate choices of function spaces, (2) construction of a class of perturbations in infinite convex optimizations germane to max–flow, and (3) a previously published classification theory of duality states defined by combinations of (extended) extremal values of closed convex bifunction dual families and their homogeneous derivant bifunctions.

This paper presents a new strongly polynomial cut canceling algorithm for minimum cost submodular flow. The algorithm is a generalization of our similar cut canceling algorithm for ordinary min-cost flow. The algorithm scales a relaxed optimality parameter and creates a second, inner relaxation that is a kind of submodular max flow problem. The outer relaxation uses a novel technique for relaxing the submodular constraints that allows our previous proof techniques to work. The algorithm uses the min cuts from the max flow subproblem as the relaxed most positive cuts it chooses to cancel. We show that this algorithm needs to cancel only ${\mathrm O}(n^3)$ cuts per scaling phase, where n is the number of nodes. Furthermore, we show how to slightly modify this algorithm to get a strongly polynomial running time. Finally, we briefly show how to extend this algorithm to the separable convex cost case and that the same technique can be used to construct a polynomial time maximum mean cut canceling algorithm for submodular flow.

The maximum flow and minimum cost-maximum flow problems are both concerned with determining flows through a network between a source and a destination. Maximum flow applies to any problem where the objective is move as many as possible goods/objects/people between two locations via intermediate locations. Maximum flow-minimum cost applies to flow problems where both capacities and costs are involved. When given information about a network (network flow diagram, capacities, costs), computing enables one to arrive at a solution to the problem. Once the solution becomes available, it has to be applied to a real world problem. The use of the following computer software in solving these problems will be discussed: R (several packages and functions), specially written Pascal programs and Excel SOLVER. The minimum cost-maximum flow solutions to the following problems will also be discussed: maximum flow, minimum cost-maximum flow, transportation problem, assignment problem, shortest path problem, caterer problem.

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.

We provide a comprehensive study on network flow problems with arc reversal capabilities. The problem is to identify the arcs to be reversed in order to achieve a maximum flow from source(s) to sink(s). The problem finds its applications in emergency transportation management, where the lanes of a road network could be reversed to enable flow in the opposite direction. We study several network flow problems with the arc reversal capability and discuss their complexity. More specifically, we discuss the polynomial time algorithms for the maximum dynamic flow problem with arc reversal capability having a single source and a single sink, and for the maximum (static) flow problem. The presented algorithms are based on graph transformations and reductions to polynomially solvable flow problems. In addition, we show that the quickest transshipment problem with arc reversal capability and the problem of minimizing the total cost resulting from arc switching costs are \(\mathcal{NP}\) -hard.

In this paper, we will extend the results about the parametric maximum flow problem to networks in which the parametrization of the arc capacities can involve both the source and the sink, as in Gallo, Grigoriadis, and Tarjan (1989), and also an additional node. We will show that the minimum cuts of the investigated networks satisfy a relaxed form of the generalized nesting property (Arai, Ueno, and Kajitani, 1993). A consequence is that the corresponding parametric maximum flow value function has at most n −1 breakpoints. All the minimum cut capacities can therefore be computed by O(1) maximum flow computations.

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

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

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.

Maximum flow problem on dynamic generative network flows with time-varying bounds

We present a new technique for determining how much information about a program's secret inputs is revealed by its public outputs. In contrast to previous techniques based on reachability from secret inputs (tainting), it achieves a more precise quantitative result by computing a maximum flow of information between the inputs and outputs. The technique uses static control-flow regions to soundly account for implicit flows via branches and pointer operations, but operates dynamically by observing one or more program executions and giving numeric flow bounds specific to them (e.g., 17 bits). The maximum flow in a network also gives a minimum cut (a set of edges that separate the secret input from the output), which can be used to efficiently check that the same policy is satisfied on future executions. We performed case studies on 5 real C, C++, and Objective C programs, 3 of which had more than 250K lines of code. The tool checked multiple security policies, including one that was violated by a previously unknown bug.

The multicommodity flow problem arises when several different commodities are transshipped from specific supply nodes to the corresponding demand nodes through the arcs of an underlying capacity network. The maximum flow over time problem concerns to maximize the sum of commodity flows in a given time horizon. It becomes the earliest arrival flow problem if it maximizes the flow at each time step. The earliest arrival transshipment problem is the one that satisfies specified supplies and demands. These flow over time problems are computationally hard. By reverting the orientation of lanes towards the demand nodes, the outbound lane capacities can be increased. We introduce a partial lane reversal approach in the class of multicommodity flow problems. Moreover, a polynomial-time algorithm for the maximum static flow problem and pseudopolynomial algorithms for the earliest arrival transshipment and maximum dynamic flow problems are presented. Also, an approximation solution to the latter problem is obtained in polynomial-time.

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.

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.).

Today there is no complex solution to the problem of calculation and optimization of production capacity, taking into account the performance of the main aggregates and the throughput of equipment of auxiliary sections in the combined production streams of metallurgical production. There is no method for finding bottlenecks in combined production streams. There is no classification of bottlenecks. The purpose of the article is to develop scientific and methodological recommendations for optimizing the enterprise (subdivision) in terms of combined production streams. Achieving the goal requires solving the following tasks: analysis of theoretical foundations and methods of calculation of production capacity; development of a formal representation of the structure of combined production streams; choice of mathematical model for optimization of production capacity. Analysis of studies and publications shows the impossibility of solving the problem of optimizing the production capacity of an enterprise without combining the formal representation of the structure of production streams with the corresponding mathematical model. It is proposed to submit the structure of combined production streams in the form of a graph-mathematical model. A network graph of combined production streams of blast furnace production was constructed. To calculate and optimize the production capacity, the problem of the maximum flow in the network, which is the task of linear programming, is used. The problem of finding the maximum flow in the grid, that is, the maximum production output of the unit to take into account the performance of the main aggregates and the capacity of the service stations, is solved. The maximum flow in the grid means the optimal value of the production capacity. Solving the dual problem of maximum flow gives the possibility of finding a minimum cut in a network, which is analogous to the bottleneck in the combined production flows. The following classification of bottlenecks in combined production streams is proposed: local, nodular, cross-sectional. The proposed methods of research and solving the formulated problems generalize and develop the well-known theoretical principles of the economy and the organization of production.