Maximum Concurrent Flow Problem Research Articles

Flowthrough Centrality: A Stable Node Centrality Measure

This paper introduces flowthrough centrality, a node centrality measure determined from the hierarchical maximum concurrent flow problem (HMCFP). Based upon the extent to which a node is acting as a hub within a network, this centrality measure is defined to be the fraction of the flow passing through the node to the total flow capacity of the node. Flowthrough centrality is compared to the commonly-used centralities of closeness centrality, betweenness centrality, and flow betweenness centrality, as well as to stable betweenness centrality to measure the stability (i.e., accuracy) of the centralities when knowledge of the network topology is incomplete or in transition. Perturbations do not alter the flowthrough centrality values of nodes that are based upon flow as much as they do other types of centrality values that are based upon geodesics. The flowthrough centrality measure overcomes the problem of overstating or understating the roles that significant actors play in social networks. The flowthrough centrality is canonical in that it is determined from a natural, realized flow universally applicable to all networks.

Concurrent Multipath Routing Optimization in Named Data Networks

To achieve maximum throughput in named data networking (NDN) with consideration of fairness among each source-sink pair, we model the routing problem in NDN as a mixed-integer maximum concurrent flow problem. Taking advantage of the smart forwarding framework in NDN, we obtain an approximate optimal routing and forwarding solution with our improved Fleischer and random rounding (FRR) algorithm. We analyze the complexity of the Fleischer algorithm more comprehensively and give the best scaling method to achieve the least iterations. In addition, we further reduce the complexity of the Fleischer algorithm so that it iterates only once, thus getting a heuristic algorithm which is called multicommodity multipath (MCMP). Using ndnSIM-based simulation, we compare our proposed routing algorithms with the existing routing algorithms, including flooding, k-shortest path, and best single path routing, and the simulation results prove that FRR and MCMP can outperform other algorithms both in throughput and latency, and it can also be concluded that the best number of paths for k-shortest path routing should be close to and less than the average node degree of the network. For studying routing optimization with caching, we conduct simulations with different in-network cache capacity and different caching policies. The simulation results show that although the LRU can achieve the best performance under the same cache capacity, the cache capacity plays a decisive role. All routing algorithms produce better performance with caching, but the FRR still achieves the best performance, and follows by MCMP.

Resource Allocation for SVC Streaming Over Cooperative Vehicular Networks

Video streaming over vehicular networks has considerable benefits for both road safety and entertainment. In this paper, the scalable video coding over cooperative vehicular networks is investigated to improve the performance of video streaming services. We formulate a joint optimization problem to answer questions about the number of video layers that should be transmitted for each vehicle, the selection of relay vehicles to assist destination vehicles, and the assignment of radio resources to direct and cooperative communications. We decouple the problem into two subproblems, i.e., the quality of service (QoS)-driven resource allocation (QoSRA) subproblem and the layer selection (LS) subproblem. We solve the QoSRA subproblem by converting it into a generalized maximum concurrent flow problem, and propose a tabu search-based metaheuristic algorithm for the LS subproblem. The simulation results verify the efficiency of the proposed scheme and provide some insight into the tradeoff between system performance and computation complexity.

Joint Channel Assignment and Power Allocation Based on Maximum Concurrent Multicommodity Flow in Cognitive Radio Networks

Emerging cognitive radio networking technology potentially provides a promising solution to the spectrum underutilization problem in wireless access. In this paper, a cross‐layer routing for secondary multihop is studied in cognitive radio network operating in television white spaces. The framework considers a joint channel, power, and routing assignment under signal to interference noise ratio (SINR) constraints. The problem is formulated as a maximum concurrent multicommodity flow problem. The goal of conducting this research is to develop a new routing protocol for the secondary multihop cognitive radio network. Therefore, the objective of this paper focuses on maximizing a flow rate scaling factor. Moreover, the paper focuses on achieving multipath routing when it is possible under SINR constraints to utilize all possible unused channels efficiently. The numerical results proved the strength of the proposed algorithm in its routing ability under the physical model of SINR, in addition to the ability of using multipath routing if there are available free channels to be used in the cognitive communication paradigm.

Efficient algorithms for the maximum concurrent flow problem

In this article, we propose a generic decomposition scheme for the maximum concurrent flow problem. This decomposition scheme encompasses many models, including, among many others, the classical path formulation and the less studied tree formulation, where the flows of commodities sharing a same source vertex are routed on a set of trees. The pricing problem for this generic model is based on shortest‐path computations. We show that the tree‐based linear programming formulation can be solved much more quickly than the path or the aggregated arc‐flow formulation. Some other decomposition schemes can lead to even faster resolution times. Finally, an efficient strongly polynomial‐time combinatorial algorithm is proposed for the single‐source case. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 65(1), 56–67. 2015

A compact linear programming formulation of the maximum concurrent flow problem

We present an alternative linear programming formulation of the maximum concurrent flow problem (MCFP) termed the triples formulation. The standard formulations in the literature are the edge‐path and node‐edge formulations, which are known to be equivalent due to the Flow Decomposition Theorem. We present algorithms for deriving a triples solution from an edge‐path solution and vice versa, and hence show that all three formulations are equivalent. Our new formulation leads to more compact linear programs than either the edge‐path or node‐path formulations. We show that the triples formulation often has half the number of rows and half the number of columns compared to the node‐edge formulation. We report computational results comparing the solution times using the three formulations and the state‐of‐the‐art linear programming solver CPLEX on a set of popular problem instances from the literature and a set of instances defined on random geometric graphs. The results indicate that the triples formulation can be solved more efficiently than the other two. We found that the CPLEX linear programming solvers solved 89% of the MCFP instances in our computational study faster with the triples formulation than it did with the other two formulations, typically two to four times faster than the node‐edge formulation when available computer memory allowed both to be solved. The triples formulation appears to be particularly well suited for problem instances defined on dense graphs; on average, CPLEX solved these types of problems in our study 10 times faster with the triples formulation. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 65(1), 68–87. 2015

A new model for multicommodity flow problems, and a strongly polynomial algorithm for single-source Maximum Concurrent Flow

In this paper, a new decomposition approach is proposed to solve large size instances of Multicommodity Flow problems. Instead of generating paths, we generate trees in a convenient way. Numerical results show that the new approach is much more efficient than the classical paths generation approach. Moreover, we propose a combinatorial polynomial-time algorithm to solve the maximum concurrent flow problem (MCF) in the single-source case.

A Mixed Integer Model for the Sparsest Cut problem

Abstract In a capacitated graph with a set of commodities, the sparsity of a cut is the ratio between the capacity of the cut and the demand of the commodities separated by the cut. The Sparsest Cut (SC) is often introduced as a weak dual of the Maximum Concurrent Flow problem (MCF). Contrarily to MCF, problem SC is, in general, NP-hard. This problem has been considerably studied, motivating the design of very elaborated approximation algorithms [Auman, Y. and Y. Rabani, Approximate min-cut max-flow theorem and approximations algorithm, SIAM Journal on Computing 27 (1998), pp. 213–238] [Gunluk, O., On the min-cut max-flow ratio for multicommodity flows, SIAMJ. of Disc. Math. 21 (2007), pp. 1–15] [N. Garg, M. Y., V. Vazirani, Approximate max-flow min-(multi)cut theorems and their applications, in: STOC'93, 1993, pp. 698–707] [S. Plotkin, E. T., Improved bounds on the max-flow min-cut ratio for multicommodity flows, in: 25th Symposium on Theory of Computing, 1993] [Tragoudas, S., Improved approximations for the min-cut max-flow ratio and the flux, Mathematical Systems Theory 29 (1996), pp. 157–167]. Somewhat surprisingly, to the best of our knowledge, problem (SC) has not been investigated with exact approaches using Mixed Integer Programming models. In this paper, we propose a formulation arising “naturally” from the dual of (MCF).

Faster approximation algorithm for generalised maximum concurrent flow problem in networks with no flow-generating cycles

This paper presents fully polynomial approximation algorithm for generalised maximum concurrent flow problem in networks with no flow-generating cycles. This paper is showed by modifying the algorithms by Fleischer and Dong. For all commodities which have a common source the new algorithm calls subroutine to find their generalised shortest paths only one time, so the running time is independent of the number of commodities k. At the same time the new algorithm applies to not only to the lossy networks but also to the networks with no flow-generating cycles.

A fast heuristic algorithm for the maximum concurrent k-splittable flow problem

In this paper, we propose a fast heuristic algorithm for the maximum concurrent k-splittable flow problem. In such an optimization problem, one is concerned with maximizing the routable demand fraction across a capacitated network, given a set of commodities and a constant k expressing the number of paths that can be used at most to route flows for each commodity. Starting from known results on the k-splittable flow problem, we design an algorithm based on a multistart randomized scheme which exploits an adapted extension of the augmenting path algorithm to produce starting solutions for our problem, which are then enhanced by means of an iterative improvement routine. The proposed algorithm has been tested on several sets of instances, and the results of an extensive experimental analysis are provided in association with a comparison to the results obtained by a different heuristic approach and an exact algorithm based on branch and bound rules.

Capacity Scaling in Ad Hoc Networks With Heterogeneous Mobile Nodes: The Super-Critical Regime

We analyze the capacity scaling laws of mobile ad hoc networks comprising heterogeneous nodes and spatial inhomogeneities. Most of previous work relies on the assumption that nodes are identical and uniformly visit the entire network space. Experimental data, however, show that the mobility pattern of individual nodes is usually restricted over the area, while the overall node density is often largely inhomogeneous due to the presence of node concentration points. In this paper we introduce a general class of mobile networks which incorporates both restricted mobility and inhomogeneous node density, and describe a methodology to compute the asymptotic throughput achievable in these networks by the store-carry-forward communication paradigm. We show how the analysis can be mapped, under mild assumptions, into a Maximum Concurrent Flow (MCF) problem over an associated Generalized Random Geometric Graph (GRGG). Moreover, we propose an asymptotically optimal scheduling and routing scheme that achieves the maximum network capacity.

The use of sparsest cuts to reveal the hierarchical community structure of social networks

We show that good community structures can be obtained by partitioning a social network in a succession of divisive sparsest cuts. A network flow algorithm based on fundamental principles of graph theory is introduced to identify the sparsest cuts and an underlying hierarchical community structure of the network via maximum concurrent flow. Matula [Matula, David W., 1985. Concurrent flow and concurrent connectivity in graphs. In: Alavi, Y., et al. (Eds.), Graph Theory and its Applications to Algorithms and Computer Science. Wiley, New York, NY, pp. 543–559.] established the maximum concurrent flow problem (MCFP), and papers on divisive vs. agglomerative average-linkage hierarchical clustering [e.g., Matula, David W., 1983. Cluster validity by concurrent chaining. In: Felsenstein, J. (Ed.), Numerical Taxonomy: Proc. of the NATO Adv. Study Inst., vol. 1. Springer-Verlag, New York, pp. 156–166 (Proceedings of NATO ASI Series G); Matula, David W., 1986. Divisive vs. agglomerative average linkage hierarchical clustering. In: Gaul, W., and Schader, M. (Eds.), Classification as a Tool of Research. Elsevier, North-Holland, Amsterdam, pp. 289–301; Thompson, Byron J., 1985. A flow rerouting algorithm for the maximum concurrent flow problem with variable capacities and demands, and its application to cluster analysis. Master Thesis. School of Engineering and Applied Science, Southern Methodist University] provide the basis for partitioning a social network by way of sparsest cuts and/or maximum concurrent flow. The MCFP extends the [Ford Jr., Lester R., Fulkerson, Delbert R., 1956. Maximal flow through a network. Canadian Journal of Mathematics 8, 399–404; Ford Jr., Lester R., Fulkerson, Delbert R., 1962. Flows in Networks. Princeton University Press, Princeton, NJ] source–sink max-flow problem to all-pairs maximum concurrent flow. The density of an ( S, T)-cut is |( S, T)\\/(| S|•| T|) where |( S, T)| is the number of edges (equivalently, links or bonds) between communities S and T with | S|•| T| being the maximum number of edges possible. The minimum density cut in the network is the sparsest cut. When the edges are weighted, the density is the average weight of the ( S, T)-cut edges, with absent edges treated as edges of zero weight. Sparsest cuts (i.e., minimum density) of the remaining components are iteratively determined via linear programming until a multipartite cut is identified that is more constraining to concurrent flow than any sparsest cut. Empirical results on real-world networks with known hierarchical structures and random networks with embedded communities are used to compare this sparsest-cut community structure criterion to the weighted Girvan–Newman community structure criterion [e.g., Newman, M.E.J., 2004. Analysis of weighted networks. Physical Review E 70, 056131] that is based on edge betweenness centrality. A new measure of accuracy M is defined to evaluate the results of the graph partitionings found by these criteria.

An exact approach for the maximum concurrent k-splittable flow problem

We consider the maximization of a multicommodity flow throughput in presence of constraints on the maximum number of paths to be used. Such an optimization problem is strongly NP-hard, and is known in the literature as the maximum routable demand fraction variant of the k-splittable flow problem. Here we propose an exact approach based on branch and bound rules and on an arc-flow mixed integer programming formulation of the problem. Computational results are provided, and a comparison with a standard commercial solver is proposed.

Lexicographically Minimum and Maximum Load Linear Programming Problems

In this paper, we introduce the lexicographically minimum load linear programming problem, and we provide a polynomial approach followed by the proof of correctness. This problem has applications in numerous areas where it is desirable to achieve an equitable distribution or sharing of resources. We consider the application of our technique to the problem of lexicographically minimum load in capacitated multicommodity networks and discuss a special nonlinear case, the so-called Kleinrock load function. We next define the lexicographically maximum load linear programming problem and deduce a similar approach. An application in the lexicographically maximum concurrent flow problem is depicted followed by a discussion on the minimum balance problem as a special case of the lexicographically maximum load problem.

A New Min‐Cut Max‐Flow Ratio for Multicommodity Flows

In this paper we present a new bound on the min‐cut max‐flow ratio for multicommodity flow problems with specified demands. For multicommodity flows, this is a generalization of the well‐known relationship between the capacity of a minimum cut and the value of the maximum flow of a single commodity flow problem. For multicommodity flows, capacity of a cut is scaled by the demand that has to cross the cut to obtain the numerator of this ratio. In the denominator, the maximum concurrent flow value is used. Currently, the best known bound for this ratio is proportional to $\log(k)$, where k is the number of origin‐destination pairs with positive demand. Our new bound is proportional to $\log(k^*)$, where $k^*$ is the cardinality of the minimum cardinality vertex cover of the demand graph. To obtain this bound, we start with a so‐called aggregated commodity formulation of the maximum concurrent flow problem with $k^*$ commodities. We also show a similar bound for the maximum multicommodity flow problem. The new bound is proportional to $\min\{\log(k^*), k^{**}\}$, where $k^{**}$ denotes the size of the of the demand graph.

Approximating Fractional Packings and Coverings in O(1/epsilon) Iterations

We adapt a method proposed by Nesterov [Math. Program. Ser. A, 103 (2005), pp. 127-152] to design an algorithm that computes $\epsilon$-optimal solutions to fractional packing problems by solving $O(\epsilon^{-1}\sqrt{Kn\ln(m)})$ separable convex quadratic programs, where n is the number of variables, m is the number of constraints, and K is the maximum number of nonzero elements in any constraint. We show that the quadratic program can be approximated to any degree of accuracy by an appropriately defined piecewise-linear program. For the special case of the maximum concurrent flow problem on a graph $G = (V,E)$ with rational capacities and demands, we obtain an algorithm that computes an $\epsilon$-optimal flow by solving shortest path problems, i.e., problems in which the number of shortest paths computed grows as $O(\epsilon^{-1} \log(\epsilon^{-1}))$ in $\epsilon$ and polynomially in the size of the problem. In contrast, previous algorithms required $\Omega(\epsilon^{-2})$ iterations. We also describe extensions to the maximum multicommodity flow problem, the pure covering problem, and mixed packing-covering problem.

A combinatorial approximation algorithm for concurrent flow problem and its application

The multi-commodity flow problem involves simultaneous shipping several different commodities from sources to sinks in a directed network with total amount of flow going through an edge limited by its capacity. The optimization version of the multi-commodity flow problem is the maximum concurrent flow problem, which finds a flow with the minimum congestion. For any positive ε, the ε-optimal concurrent flow problem is to find a solution whose the congestion value is no more than (1+ ε) times the minimum congestion. In recent years, a few fast combinatorial approximation algorithms for the ε-optimal concurrent flow problem have been presented. In this paper we propose a new variant of the combinatorial approximation algorithm: CACF with a tighter computation bound in decreasing the values of congestion and the potential function. Numerical comparisons are made between the results obtained by the combinatorial approximation algorithms and those did by the linear programming package CPLEX on large-scale test networks. The application of CACF to efficiently solving the system-optimal network flow problem is given where good results have been obtained. It has shown the capacity of the CACF in dealing with problems of concurrent flow associated.

Asymptotic analysis of the flow deviation method for the maximum concurrent flow problem

We analyze the asymptotic behavior of the Flow Deviation Method, first presented in 1971 by Fratta, Gerla and Kleinrock, and show that when applied to packing linear programs such as the maximum concurrent flow problem, it yields a fully polynomial-time approximation scheme.

Fast deterministic approximation for the multicommodity flow problem

In this paper we consider an optimization version of the multicommodity flow problem which is known as the maximum concurrent flow problem. We show that an approximate solution to this problem can be computed deterministically using O(k(e−2 + logk) logn) 1-commodity minimum-cost flow computations, wherek is the number of commodities,n is the number of nodes, ande is the desired precision. We obtain this bound by proving that in the randomized algorithm developed by Leighton et al. (1995) the random selection of commodities can be replaced by the deterministic round-robin without increasing the total running time. Our bound significantly improves the previously known deterministic upper bounds and matches the best known randomized upper bound for the approximation concurrent flow problem.

The maximum concurrent flow problem

The maximum concurrent flow problem (MCFP) is a multicommodity flow problem in which every pair of entities can send and receive flow concurrently. The ratio of the flow supplied between a pair of entities to the predefined demand for that pair is called throughput and must be the same for all pairs of entities for a concurrent flow. The MCFP objective is to maximize the throughput, subject to the capacity constraints. We develop a fully polynomial-time approximation scheme for the MCFP for the case of arbitrary demands and uniform capacity. Computational results are presented. It is shown that the problem of associating costs (distances) to the edges so as to maximize the minimum-cost of routing the concurrent flow is the dual of the MCFP. A path-cut type duality theorem to expose the combinatorial structure of the MCFP is also derived. Our duality theorems are proved constructively for arbitrary demands and uniform capacity using the algorithm. Applications include packet-switched networks [1, 4, 8], and cluster analysis [16].