Polymatroidal flows with lower bounds

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

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

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.

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

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.

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 aim is to give rigorous formulations of max-flow problems and min-cut problems an a bounded domain Ω in R n with Lipschitz boundary and to prove max-flow min-cut theorems. For example, one has to give a rigorous notion of a flow σ, the outer unit normal v, the inner product σ.v and the cut capacity of a cut. To do so, one basically follows Strang's idea. The space of essentially bounded vector fields with divergence in L n (Ω) and the space of functions of bounded variations will play important roles in this study. In fact, a static flow is represented by an element in the former space and a static cut is represented by a characteristic function which belongs to the latter space. One of the mathematical tools is a generalized Greens' formula, for functions of bounded variation and essentially bounded vector fields with divergence in L n (Ω). Furthermore one treats the general case when the network is anisotropic

This paper aims to optimize the content-aware prioritization of scalable video multicast, which is coupled with multipath streaming and network coding based routing. It constructs multiple layer distribution meshes for the scalable video stream to minimize the total video distortion at all the receivers, determines the base layer meshes with minimum costs to maintain application-layer QoS and the layer synchronization of SVC streaming, and improves the network throughput by encouraging path-overlapping transmissions and thus allowing bandwidth sharing among different receivers for the same video layer by utilizing network coding. The targeted problem is formulated into a minimization programming in which the quality variation between layers, the transmission cost of the base layer, as well as the efficient resource utilization are jointly considered. By decomposition and dual approach, the global convex problem is solved by a two-level decentralized iterative algorithm. The implementation of the distributed algorithm is discussed with regard to the communication overhead, and the convergence performance is validated by numerical experiments. Packet-level simulations demonstrate that the proposed algorithm could approximately achieve the maximum flow rates determined by Max-Flow Min-Cut Theorem and benefit the overall received video quality.

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.

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.

Max-flow min-cut theorem

The literature of combinatorial optimization, and of related fields such as graph theory. is replete with numerous examples of duality theorems and their applications: the Max-Flow Min-Cut Theorem, Konig-Egervary Theorem, Edmond’s Odd Set Covering Theorem, Hoffman’s Circulation Theorem, Minty’s Painting Theorem, Menger’s Theorem, Dilworth’s Theorem, etc. Our objective in this Chapter is to review a few of these duality results and to give some examples which may be convincing of the usefulness of duality concepts in problem formulation and solution.

This paper addresses the clinically challenging problem of hairline mandibular fracture detection from a sequence of Computed Tomography (CT) images. A hairline fracture of critical clinical importance, can be easily missed due to the absence of sharp surface and contour discontinuities and the presence of intensity inhomogeneity in the CT image, if not scrutinized carefully. In this work, the 2D CT image slices displaying a mandible with hairline fractures are first identified within an input CT image sequence of a fractured craniofacial skeleton. This is achieved via an intensity-based image retrieval scheme using Kolmogorov-Smirnov distance as the measure of similarity and an unbroken mandible as the reference image. Since a hairline fracture is essentially a discontinuity in the bone contour, we model it as a minimum cut in an appropriately weighted flow network. The existing graph cut-based segmentation schemes are enhanced with a novel construction of the flow network, guided by the geometry of the human mandible. The Edmonds-Karp refinement of the classical Ford-Fulkerson algorithm is employed next to obtain a minimum cut, which represents the hairline fracture in the already identified CT image slices. Experimental results demonstrate the effectiveness of the proposed method.

This paper applies network flow method, genetic algorithms, and Monte Carlo simulation to optimal reliability design for a composite electric power system. Genetic algorithms are general purpose optimization techniques based on principles inspired from the biological evolution using three main operations of reproduction, crossover, and mutation, which could locate near optimal solutions in most cases. The proposed method primarily adopted Monte Carlo simulation method, maximum-flow minimum-cut theorem, and optimization techniques to find out the optimal values of reliability indices, such that the optimal reliability design for the system can be achieved. The objective function to be optimized is composed of interruption cost and installation cost. The reliability indices mainly used include expected demand not served (EDNS) and forced outage rate (FOR). An application of the proposed method conducted on an IEEE five-bus test system is presented.

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.