Maximum Multicommodity Flow Research Articles

Ensuring sustainable strategies for achieving multi-commodity maximum flow on a fuzzy network under interdictions

Ensuring sustainable strategies for achieving multi-commodity maximum flow on a fuzzy network under interdictions

Angels against demons: Fight against smuggling in an illicit supply chain with uncertain outcomes and unknown structure

Illegal smuggling is one of the main concerns in many countries due to its significant negative economic and societal impacts. To control the problem, law enforcement officials work to prevent illegal smuggling operations. However, these officials usually encounter a multi-commodity network with an unidentified structure where traffickers trade illicit commodities at different levels. Therefore, they should manage their resources to prioritize identifying the network members, monitoring the identified members, and arresting them to remove their corresponding illegal trades from the network. To manage these operations, this study aims to utilize a mathematical approach, including a formulation and solution method, to manage the available resources in the network and minimize the illicit flows between the origins and destinations. The proposed approach considers real-world conditions, including multi-commodity flows and operations uncertainty. In other words, we investigate a multi-commodity maximum flow network interdiction problem with uncertain outcomes and an unknown network structure. The network under investigation denotes the relationships among criminals who trade illegal commodities in the distribution network. Given the complexity of the problem, this study contributes to the literature by developing a bi-level mixed-integer programming formulation to interdict maximum flow in a multi-commodity, multi-source relationship network considering uncertain outcomes of leader’s attacks. In addition, it develops a solution procedure based on the Benders decomposition algorithm to solve the developed formulation. Finally, to show the practicality of the developed formulation and solution method in real-world problems and to demonstrate their performance, we conduct a simulation analysis to determine how considering uncertainty affects the decisions. Our obtained results reveal that although considering the outcome uncertainty increases the solution time, it also increases the number of arrested criminals by about 29% on average and decreases the volume of trafficking by up to 14%.

Experimental Evaluation of Algorithms for Packet Routing in Software Defined Network

The packet routing problem in a software defined network is formulated and a greedy algorithm is proposed. An alternative approach to approximately solve this problem by means of reduction to a special case of the fractional length-bounded maximum multicommodity flow is suggested. Experimental comparison of the greedy algorithm and a fully polynomial-time approximation scheme (FPTAS) for fractional length-bounded maximum multicommodity flow is carried out on the testing instances representing prospective software defined satellite network.

Maximum Multicommodity Flow with Intermediate Storage

The multicommodity flow problem deals with the transshipment of more than one commodity from respective sources to corresponding sinks without violating the capacity constraints. Due to the capacity constraints, flows out from the sources may not reach their sinks, and so, the storage of excess flows at intermediate nodes plays an important role in the maximization of flow values. In this paper, we introduce the maximum static as well as maximum dynamic multicommodity flow problems with intermediate storage. We present polynomial and pseudopolynomial time algorithms for the former and latter problems, respectively. We also present the solution procedures to these problems in contraflow network having symmetric as well as asymmetric arc transit times. We transform the solutions in continuous-time settings by using natural transformation.

On three approaches to length-bounded maximum multicommodity flow with unit edge-lengths

The paper presents a comparison between three approaches to solving the length-bounded maximum multicommodity flow problem with unit edge-lengths. Following the first approach, Garg and K?nemann?s, we developed an improved fully polynomial time approximation scheme for this problem. As the second alternative, we considered the well-known greedy approach. The third approach is the one that yields exact solutions by means of a standard LP solver applied to an LP model on the time-expanded network. Computational experiments are carried out on benchmark graphs and the graphs that model software defined satellite networks, to compare the proposed algorithms with an exact linear programming solver. The results of the experiments demonstrate a trade-off between the computing time and the precision of algorithms under consideration.

Topology Abstraction Service for IP VPNs: Core Network Partitioning for Resource Sharing

VPN service providers (VSP) and IP-VPN customers have traditionally maintained service demarcation boundaries between their routing and signaling entities. This has resulted in the VPNs viewing the VSP network as an opaque entity and therefore limiting any meaningful interaction between the VSP and the VPNs. A key challenge is to expose each VPN to information about available network resources through an abstraction (TA) [1] which is both accurate and fair. In [2] we proposed three decentralized schemes assuming that all the border nodes performing the abstraction have access to the entire core network topology. This assumption likely leads to over- or under-subscription. In this paper we develop centralized schemes to partition the core network capacities, and assign each partition to a specific VPN for applying the decentralized abstraction schemes presented in [2]. First, we present two schemes based on the maximum concurrent flow and the maximum multicommodity flow (MMCF) formulations. We then propose approaches to address the fairness concerns that arise when MMCF formulation is used. We present results based on extensive simulations on several topologies, and provide a comparative evaluation of the different schemes in terms of abstraction efficiency, fairness to VPNs and call performance characteristics achieved.

Capacity Augmentation in Wireless Mesh Networks

In this paper, we address the following problem: Given a specific deployment of a wireless mesh network and given a budget to augment the capacity of N links within the network, which links should be selected for capacity augmentation in order to maximize the achievable aggregate throughput for all the flows in the network? Unlike previous research which focuses on installing additional nodes or gateways, we work on an existing deployment of the mesh network and identify optimal link positions for capacity augmentation. We model the capacity augmentation problem in single-radio single-channel mesh networks as a multicommodity maximum flow problem and derive a mixed-integer linear programming model for identifying the optimal links for capacity augmentation. For ensuring fairness among flows, we propose a concurrent flow problem formulation and derive a corresponding mixed-integer linear programming model. We propose greedy algorithms to solve the mixed-integer linear programming formulations. We also extend our analysis to the case of multi-radio multi-channel networks in which additional radios can be deployed on nodes for capacity augmentation. The simulation results of the proposed schemes for several different topologies indicate that the proposed greedy algorithms are correct and provide exact results at a fraction of the cost.

Max-Flow Min-Cost Routing in a Future-Internet with Improved QoS Guarantees

A Constrained Multicommodity Maximum-Flow-Minimum-Cost routing algorithm is presented. The algorithm computes maximum-flow routings for all smooth unicast traffic demands within the Capacity Region of a network subject to routing cost constraints. The edge cost can be a distance, reliability, congestion or an energy metric. It is shown that every network has a finite Bandwidth-Cost capacity. The Bandwidth-Distance and the Bandwidth-Energy capacities are explored. The routing algorithm requires the formulation of two Linear Programs (LPs). The first LP finds a multicommodity Maximum-Flow, when the flows are constrained to a sub-graph of the network to enforce cost constraints. The second LP minimizes the routing cost, given that the maximum-flow is fixed. A related Constrained Multicast-Max-Flow-Min-Cost algorithm is also presented, to maximize the throughput of a multicast tree using network coding, subject to routing cost constraints. These algorithms have polynomial-time solutions, whereas traditional multipath routing algorithms can be NP-Hard. The addition of routing cost constraints can significantly reduce the size of the LPs, resulting in faster solutions, with lower edge utilizations and with higher energy efficiencies. The application of these algorithms to route aggregated video streams from cloud data centers in a Future-Internet network, with improved throughput, energy-efficiency and QoS guarantees is presented.

Analyzing quadratic unconstrained binary optimization problems via multicommodity flows

Quadratic Unconstrained Binary Optimization (QUBO) problems concern the minimization of quadratic polynomials in n { 0 , 1 } -valued variables. These problems are NP-complete, but prior work has identified a sequence of polynomial-time computable lower bounds on the minimum value, denoted by C 2 , C 3 , C 4 , … . It is known that C 2 can be computed by solving a maximum flow problem, whereas the only previously known algorithms for computing C k ( k > 2 ) require solving a linear program. In this paper we prove that C 3 can be computed by solving a maximum multicommodity flow problem in a graph constructed from the quadratic function. In addition to providing a lower bound on the minimum value of the quadratic function on { 0 , 1 } n , this multicommodity flow problem also provides some information about the coordinates of the point where this minimum is achieved. By looking at the edges that are never saturated in any maximum multicommodity flow, we can identify relational persistencies: pairs of variables that must have the same or different values in any minimizing assignment. We furthermore show that all of these persistencies can be detected by solving single-commodity flow problems in the same network.

Polynomial flow-cut gaps and hardness of directed cut problems

We study the multicut and the sparsest cut problems in directed graphs. In the multicut problem, we are a given ann-vertex graphGalong withksource-sink pairs, and the goal is to find the minimum cardinality subset of edges whose removal separates all source-sink pairs. The sparsest cut problem has the same input, but the goal is to find a subset of edges to delete so as to minimize the ratio of the number of deleted edges to the number of source-sink pairs that are separated by this deletion. The natural linear programming relaxation for multicut corresponds, by LP-duality, to the well-studied maximum (fractional) multicommodity flow problem, while the standard LP-relaxation for sparsest cut corresponds to maximum concurrent flow. Therefore, the integrality gap of the linear programming relaxation for multicut/sparsest cut is also theflow-cut gap: the largest gap, achievable for any graph, between the maximum flow value and the minimum cost solution for the corresponding cut problem.Our first result is that the flow-cut gap between maximum multicommodity flow and minimum multicut is Ω˜(n1/7) in directed graphs. We show a similar result for the gap between maximum concurrent flow and sparsest cut in directed graphs. These results improve upon a long-standing lower bound of Ω(logn) for both types of flow-cut gaps. We notice that these polynomially large flow-cut gaps are in a sharp contrast to the undirected setting where both these flow-cut gaps are known to be Θ(logn). Our second result is that both directed multicut and sparsest cut are hard to approximate to within a factor of 2Ω(log1-ϵn)for any constant ϵ > 0, unless NP ⊆ ZPP. This improves upon the recent Ω(logn/log logn)-hardness result for these problems. We also show that existence of PCP's for NP with perfect completeness, polynomially small soundness, and constant number of queries would imply a polynomial factor hardness of approximation for both these problems. All our results hold for directed acyclic graphs.

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.

Dynamic online QoS routing schemes: Performance and bounds

Several dynamic QoS routing techniques have been recently proposed for new IP networks based on label forwarding. However, no extensive performance evaluation and comparison is available in the literature. In this paper, after a short review of the major dynamic QoS routing schemes, we analyze and compare their performance referring to several networks scenarios. In order to set an absolute evaluation of the performance quality we have obtained the ideal performance of any routing scheme using a novel and flexible mathematical programming model that assumes the knowledge of arrival times and duration of the connections offered to the network. This model is based on an extension of the maximum multi-commodity flow problem. Being an integer linear programming model, its complexity is quite high and its evaluation is constrained to networks of limited size. To overcome the computational complexity we have defined an approximate model, based on the multi-class Erlang formula and the minimum multi-commodity cut problem, that provides an upper bound to the routing scheme performance. The performance presented in the paper, evaluated by measuring the connection rejection probability, shows that the schemes considered reach, in several scenarios, the ideal performance, showing that no much gain is left for alternate new schemes.

A Lower Bound On The Integrality Gap For Minimum Multicut In Directed Networks

Given a directed edge-weighted graph and k source-sink pairs, the Minimum Directed Multicut Problem is to find an edge subset with minimal weight, that separates each source-sink pair. Determining the minimum multicut in directed or undirected graphs is NP-hard. The fractional version of the minimum multicut problem is dual to the maximum multicommodity flow problem. The integrality gap for an instance of this problem is the ratio of the minimum weight multicut to the minimum weight fractional multicut; trivially this gap is always at least 1 and it is easy to show that it is at most k. In the analogous problem for undirected graphs this upper bound was improved to O(log k).In this paper, for each k an explicit family of examples is presented each with k source-sink pairs for which the integrality gap can be made arbitrarily close to k. This shows that for directed graphs, the trivial upper bound of k can not be improved.

Two-commodity reliability evaluation for a stochastic-flow network with node failure

This paper considers a stochastic-flow network, in which each node as well as each arc has several capacities and may fail, which allows each commodity to be transmitted from the source to the sink. Given the demand for each commodity, the evaluation of the system reliability to fulfill the demand is performed. At first, a simple algorithm is proposed to generate all lower boundary points for the demand, and then the system reliability can be calculated in terms of such points. An example is shown to illustrate the solution procedure. Scope and purpose Assuming that the arcs of a flow network are deterministic, the multicommodity maximum flow (MMF) problem is a standard one in network analysis and operations research. The purpose of this paper is to extend the MMF problem to a stochastic-flow network in which arcs and nodes all have several capacities and may fail. We evaluate the probability (named system reliability here) that the system satisfies the demand for each commodity simultaneously. Such a probability can be treated as a performance index for many supply–demand systems such as computer systems, telecommunication systems, logistics systems, etc.

Study on the multicommodity reliability of a capacitated-flow network

Traditionally, many researchers solved the multicommodity maximum flow problem by assuming that the arcs of the flow network are deterministic. When the arcs are stochastic (i.e., the capacity of each arc has several values), this article studies how to calculate the probability that a capacitated-flow network with a unique source mode satisfies a demand ( d 1, d 2,…, d p ) at the unique sink node, where d k is the demand of commodity k. Such a probability is named the multicommodity reliability and is dependent on capacities of arcs. One solution procedure is proposed to evaluate the multicommodity reliability, which includes two parts: an algorithm to generate all ( d 1 , d 2 ,…, d p )-MPs and a method to calculate the multicommodity reliability in terms of ( d 1, d 2,…, d p )-MPs. Two illustrative examples are given.

Combinatorial Approximation Algorithms for Generalized Flow Problems

Generalized network flow problems generalize normal network flow problems by specifying a flow multiplier μ((v,w)) for each arc (v,w). For every unit of flow entering the arc, μ((v,w)) units of flow exit. We present a strongly polynomial algorithm for a single-source generalized shortest paths problem, using a left-distributive closed semiring. This permits use of a dynamic-programming routine similar to the Bellman–Ford algorithm, given a guess for the value of the optimal solution. Using Megiddo's parametric search scheme, we can compute the optimal value in strongly polynomial time. The algorithm's running time O(mn2logn) matches the previously best known, but the algorithm is simpler, is based on the well-known theory of closed semirings, and directly works with the given graph. All previous polynomial-time algorithms were based on interior-point methods or directly solved the dual problem and translated the solution back to the primal problem. Using this generalized shortest paths algorithm, we present fully polynomial-time approximation schemes for the generalized versions of the maximum flow, the nonnegative-cost minimum-cost flow, the concurrent flow, the multicommodity maximum flow, and the multicommodity nonnegative-cost minimum-cost flow problems with running times independent of the size of the flow multipliers' representation.

A highly efficient path-restoration protocol for management of optical network transport integrity

Distributed path restoration based on optical cross-connects can provide highly capacity-efficient real-time restoration for WDM-based optical networking. However, to obtain an assured restoration level with the theoretically very low amounts of spare that path restoration allows, one must solve, or closely approximate a solution to, the integer multicommodity maximum flow (MCMF) problem, MCMF is, however a hard combinatorial optimization problem due to what is called the mutual capacity aspects of the problem: which of many competing origin-destination pairs should be allowed paths over the finite spares on each span? Integer MCMF is further complicated by the nonunimodular nature of the problem, i.e., fractional flows are forbidden but would arise if solved by linear programming. This paper presents a heuristic principle that tests well against integer programming solutions of MCMF routing. The heuristic is first characterized in a centralized program, then adapted for use in a distributed path restoration protocol. In all test cases, the protocol obtains over 97% of the paths found in an optimal MCMF solution in the same network. Via OPNET simulation it is also predicted that the protocol will run in well under 2 seconds which means it could be used directly in real-time, or in distributed prefailure self-planning, for restoration. The significance is that network operators could aggressively optimize their spare capacity, toward theoretical minimums, while still assuring 100% restorability.

Approximating Fractional Multicommodity Flow Independent of the Number of Commodities

We describe fully polynomial time approximation schemes for various multicommodity flow problems in graphs with m edges and n vertices. We present the first approximation scheme for maximum multicommodity flow that is independent of the number of commodities k, and our algorithm improves upon the run time of previous algorithms by this factor of k, running in ${{\cal O}^*(\epsilon^{-2}m^2)}$ time. For maximum concurrent flow and minimum cost concurrent flow, we present algorithms that are faster than the current known algorithms when the graph is sparse or the number of commodities k is large, i.e., k > m/n. Our algorithms build on the framework proposed by Garg and Könemann [Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, IEEE, New York, 1998, pp. 300--309]. They are simple, deterministic, and for the versions without costs, they are strongly polynomial. The approximation guarantees are obtained by comparison with dual feasible solutions found by our algorithm. Our maximum multicommodity flow algorithm extends to an approximation scheme for the maximum weighted multicommodity flow, which is faster than those implied by previous algorithms by a factor of k/log W, where W is the maximum weight of a commodity.

Fast and efficient linear programming and linear least-squares computations

We present a new parallel algorithm for computing a least-squares solution to a sparse overdetermined system of linear equations A x = b such that the m × n matrix A is sparse and the graph, G = ( V, E), of the matrix H= I A T A O has an s( m + n)-separator family, i.e. either | V| < n 0 for a fixed constant n 0 or, by deleting a separator subset S of vertices of size ⩽ s( m + n), G can be partitioned into two disconnected subgraphs having vertex sets V 1, V 2 of size ⩽ 2 3 (m + n) , and each of the two resulting subgraphs induced by the vertex sets S U ́ V i, i = 1, 2 , can be recursively s(|S U ́ V i|) -separated in a similar way. Our algorithm uses O(log ( m + n) log 2 s ( m + n)) steps and ⩽ s 3( m + n) processors; it relies on our recent parallel algorithm for solving sparse linear systems and has several immediate applications of interest, in particular to mathematical programming, to sparse nonsymmetric systems of linear equations and to the path algebra computations. We most closely examine the impact on the linear programming problem (LPP) which requires maximizing c T y subject to A T y ⩽ b, y ⩾ 0 , where A is an m × n matrix. Hereafter it is assumed that m ⩾ n. The recent algorithm by Karmarkar gives the best-known upper estimate [ O ( m 3.5 L) arithmetic operations, where L is the input size] for the cost of the solution of this problem in the worst case. We prove an asymptotic improvement of that result in the case where the graph of the associated matrix H has an s ( m + n)-separator family; then our algorithm can be implemented using O ( mL log m log 2 s ( m + n)) parallel arithmetic steps, s 3 ( m + n) processors and a total of O ( mLs 3 ( m + n) log m log 2 s ( m + n)) arithmetic operations. In many cases of practical importance this is a considerable improvement on the known estimates: for example, s ( m + n) = √8 ( m + n) if G is planar [as occurs in many operations research applications; for instance, in the problem of computing the maximum multicommodity flow with a bounded number of commodities in a network having an s ( m + n)-separator family], so that the processor bound is only 8 √8 ( m + n) 1.5 and the total number of arithmetic steps is O ( m 2.5 L). Similarly, Karmakar's algorithm and the known algorithms for the solution of overdetermined linear systems are accelerated in the case of dense input matrices via our recent parallel algorithms for the inversion of dense k × k matrices using O (log 2 k) steps and k 3 processors. Combined with a modification of Karmarkar's algorithm, this implies solution of the LPP using O ( Lm log 2 m) steps and m 2.5 processors. The stated results promise some important practical applications. Theoretically, the above processor bounds can be reduced for dense matrix inversion to o ( k 2.5) and for the LPP to o ( m 2.165) in the dense case and to o ( s 2.5 ( m + n)) in the sparse case (preserving the same number of parallel steps); this also decreases the known sequential time bound for the LPP by a factor of m 0.335, i.e. to O ( Lm 3.165).