Maximum Flow In Network Research Articles

Quick Max-flow Algorithm

Determining of a maximal network flow is a classic problem in discrete optimization with many applications. In this paper, a new algorithm based on the Dinic’s method is presented. Algorithms of the Dinic’s method work evidently faster than theoretical bounds for a randomized network. This paper presents a parameterized and easy to implement family of algorithms of finding a saturating flow in a layered network. Although their common complexity is poor O(V2L) where L is the number of layers, three particular members are proved to be O(V2). Furthermore, there is a particularly interesting “balanced” member of the family for which a calculated upper bound on complexity is still O(V2L) but there is known no example of a layered network that needs more than O(E + V(3/2)) time to resolve. All the considered members work really quickly for randomized examples of a layered network. Starting from the above family, three algorithms which find maximal flow in a network in O(V3) worst case time have been constructed, while the respective “balanced” algorithm is theoretically O(V4). All the algorithms do not extend O(V2) time in experimental, i.e. randomized, cases.

Rainbow Network Flow of Multiple Description Codes

This paper is an enquiry into the interaction between multiple description coding (MDC) and network routing. We are mainly concerned with rate-distortion optimized network flow of a multiple description (MD) source from multiple servers to multiple sinks. We aim at maximizing a collective metric of the quality of source reconstruction at all sinks, by optimally routing the MD source streams from the server nodes to the sinks. This problem turns out to be very different from conventional maximum network flow. The objective function involves not only the flow volume but also the diversity of the flow contents (i.e., distinction of descriptions), hence, the term rainbow network flow (RNF). For a general network topology, a general fidelity function, and an arbitrary distribution of MDC descriptions on the servers, we prove the RNF problem to be Max-SNP-hard. However, the problem becomes tractable in many practical scenarios, such as when MDC is balanced with descriptions of the same length and importance, when all source nodes have the complete set of MDC descriptions, and when the network topology is a tree or has only one sink. Polynomial-time RNF algorithms are developed for these cases.

Network Coding for Efficient Multicast Routing in Wireless Ad-hoc Networks

Network coding is a powerful coding technique that has been proved to be very effective in achieving the maximum multicast capacity. It is especially suited for new emerging networks such as ad-hoc and sensor networks. In this work, we investigate the multicast routing problem based on network coding and put forward a practical algorithm to obtain the maximum flow multicast routes in ad-hoc networks. The "conflict phenomenon" that occurs in undirected graphs will also be discussed. Given the developed routing algorithm, we will present the condition for a node to be an encoding node along with a corresponding capacity allocation scheme. We will also analyze the statistical characteristics of encoding nodes and maximum flow in ad-hoc networks based on random graph theory.

Capacity analysis of a towline conveyor system in a distribution center

In a materials flow system, it is important to know the system loading capacity and identify bottleneck links in a short-or long-term planning horizon; and it is beneficial to optimise workforce allocation in order to achieve the best possible system performance under static or dynamic conditions. A multi-commodity maximum network flow model is developed in this paper to determine the flow capacity in a towline conveyor system and to optimise workforce allocation in the system. The proposed modelling approach in this paper uses three types of nodes and two types of arcs to address various types of linkages and processes within a towline conveyor network in a distribution centre. Multiple flow layers are applied to consider various flow paths and commodities. The proposed model and modelling approach can assist in capacity analysis and staffing allocation for a large-scale towline conveyor system in a distribution centre. An industrial case study is presented to illustrate the application of the modelling and analysis approach to determine the maximal flow of a towline conveyor system and identify improvement opportunities.

Maximum flow and network capacity of network coding for ad-hoc networks

Network coding is an effective way to achieve the maximum flow of multicast networks. In this letter, we focus on the statistical properties of the maximum flow or the capacity of network coding for ad-hoc networks based on random graph models. Theoretical analysis shows that the maximum flow can be modelled as extreme order statistics of Gaussian distribution for both wired and wireless ad-hoc networks as the node number is relatively large under a certain condition. We also investigate the effects of the nodes' covering capabilities on the capacity of network coding.

Memetic Algorithms: The Polynomial Local Search Complexity Theory Perspective

In previous work (Krasnogor, http://www.cs.nott.ac.uk/~nxk/papers.html. In: Studies on the Theory and Design Space of Memetic Algorithms. Ph.D. thesis, University of the West of England, Bristol, UK, 2002; Krasnogor and Smith, IEEE Trans Evol Algorithms 9(6):474–488, 2005) we develop a syntax-only classification of evolutionary algorithms, in particular so-called memetic algorithms (MAs). When “syntactic sugar” is added to our model, we are able to investigate the polynomial local search (PLS) complexity of memetic algorithms. In this paper we show the PLS-completeness of whole classes of problems that occur when memetic algorithms are applied to the travelling salesman problem using a range of mutation, crossover and local search operators. Our PLS-completeness results shed light on the worst case behaviour that can be expected of a memetic algorithm under these circumstances. Moreover, we point out in this paper that memetic algorithms for graph partitioning and maximum network flow (both with important practical applications) also give rise to PLS-complete problems.

Model to Forecast Maximum Flows in On-Demand Irrigation Distribution Networks

A model is developed using data from an on-demand pressurized water distribution network located in sector VIII of the Genil-Cabra irrigation district of Santaella, Cordoba (southern Spain), to simulate an irrigation season and calculate the flows that circulate in the system at any given time during the irrigation day. Using the results obtained by the model, water demand frequencies can be estimated. These results are compared to those attained by Clement’s and Mavropoulos’s formulas. This procedure enables us to determine to what extent it is possible to adjust statistical distributions to the demand obtained by both models and to verify if the hypotheses upon which these models are based are, in fact, fulfilled. Moreover, we are able to study which periods are the most appropriate for determining peak demand. Our results show that the statistical methods slightly underestimate demand because demand tends to be concentrated at two peak times during the day: one at mid-morning and another in the late af...

Achieving maximum flow in interference-aware wireless sensor networks with smart antennas

Directional antenna offers various benefits for wireless sensor networks, such as increased spatial reuse ratio and reduced energy consumption. In this paper, we formulate the maximum flow problem as an optimization problem in interference-limited wireless sensor networks with switched beam directional antennas. The optimization problem is solvable in the presence of an omniscient controller, but it is NP-hard. Therefore, we seek a distributed algorithm to achieve the maximum flow through jointly routing and scheduling. The maximum flow between given source destination pair is determined forwardly hop by hop and is verified by the proposed feasible condition at downstream nodes. This method works for both single-beam antenna and multi-beam antenna with some variation in the feasibility condition.

Matrix games in the multicast networks: maximum information flows with network switching

Network coding for achieving the maximum information flow in the multicast networks has been proposed by Ahlswede, Cai, Li, and Yeung. They have demonstrated that the conventional network switching, without resort to network coding, is in general not able to achieve the optimum information flow that has been promised by network coding. A basic problem arising here is that, for a given multicast network, what is the switching gap of the network defined as the ratio of the maximum information flow in the multicast network with network coding to that only with network switching. In the paper, by considering network switching as a special form of network coding, we make a complete theoretical and computational determination of the achievable information rate region for multisource multicast network switching. The multicast networks are allowed to be cyclic or acyclic with links having arbitrary positive integer or real-valued capacity. Network switching is essentially a problem of multicast-route packing in the multicast networks. Based on this, we use the theory of games to formulate the network switching as a matrix game between the first player of links and the second player of multicast routes in the multicast networks. We prove that the maximum achievable information rate at each probabilistic direction in the information rate region is the reciprocal of the value of the corresponding game. Consequently, the maximum achievable information rate can be computed in a simple way by applying the existing theory and algorithms for the computation of the value of a matrix game, especially for such multicast networks with links all having unit capacity as the Ahlswede-Cai-Li-Yeung multicast networks. For multicast networks with links having arbitrary positive real-valued capacity, by using convex optimization, we develop a simple and efficient iterative algorithm to find the maximum achievable information rates for multisource multicast network switching. For single-source multicast networks whose links have arbitrary positive real-valued capacity, we present two max-flow min-cut theorems. The maximum information flow for single-source network switching is the minimum capacity among all soft link-cuts of the multicast network, while the maximum information flow for single-source network coding is the minimum capacity among all hard link-cuts. Consequently, by applying the theory of approximation algorithms for the set-covering problem, we demonstrate that the switching gap of a single-source multicast network is upper-bounded by the nth harmonic number Hn=1+1/2+...+1/n where n is the largest number of multicast routes containing a given link in the network. This harmonic-number bound is asymptotically tight as Oscr(ln n) for the combination multicast network. For the special class of multisource multicast networks with the same set of sink nodes, we make a comparison between the achievable information rate regions for network switching and network coding

Embedding k-Outerplanar Graphs into l1

We show that the shortest-path metric of any $k$-outerplanar graph, for any fixed $k$, can be approximated by a probability distribution over tree metrics with constant distortion and hence also embedded into $\ell_1$ with constant distortion. These graphs play a central role in polynomial time approximation schemes for many NP-hard optimization problems on general planar graphs and include the family of weighted $k\times n$ planar grids. This result implies a constant upper bound on the ratio between the sparsest cut and the maximum concurrent flow in multicommodity networks for $k$-outerplanar graphs, thus extending a theorem of Okamura and Seymour [J. Combin. Theory Ser. B, 31 (1981), pp. 75-81] for outerplanar graphs, and a result of Gupta et al. [Combinatorica, 24 (2004), pp. 233-269] for treewidth-2 graphs. In addition, we obtain improved approximation ratios for $k$-outerplanar graphs on various problems for which approximation algorithms are based on probabilistic tree embeddings. We conjecture that these embeddings for $k$-outerplanar graphs may serve as building blocks for $\ell_1$ embeddings of more general metrics.

A graph-oriented approach for the minimization of the number of late jobs for the parallel machines scheduling problem

An O( n 2) algorithm is presented for the n jobs m parallel machines problem with identical processing times. Due dates for each job are given and the objective is the minimization of the number of late jobs. Preemption is permitted. The problem can be formulated as a maximum flow network model. The optimality proof as well as other properties and a complete example are given.

The symbolic algorithms for maximum flow in networks

The maximum flow problem is one of the classic combinatorial optimization problems with many applications, such as electrical powers, traffics, communications, computer networks and logistics. The problem is to find a flow of maximum value on a network from a source to a sink. Ordered binary decision diagram (OBDD) is a canonical form to represent and manipulate the Boolean functions efficiently. OBDD-based symbolic algorithms appear to give improved results for large-scale combinatorial optimization problems by searching the nodes and edges implicitly. For the maximum flow problem in networks, we present the symbolic algebraic decision diagram (ADD) formulation and symbolic algorithms. The augmenting-path-based symbolic algorithm is the combination of the Gabow's scaling algorithm with the binary decision diagram (BDD)-based symbolic algorithm for the maximum flow in 0–1 networks. The Karzanov's algorithm is implemented implicitly, resulting in the preflow-based symbolic algorithm (PSA), in which the vertices on each layer of the layered networks are partitioned by the vertex's input preflow and the total capacity of its outgoing edges, and the edges to push or pull the preflow are selected in terms of the priority function. The improved PSA is developed by integrating the heuristic of Träff's algorithm and the assistant layered networks into the PSA. The symbolic algorithms do not require explicit enumeration of the nodes and edges, and therefore can handle large-scale networks.

Algorithm model research of the logical cutting tree on the network maximum flow

It is the complexity and the calculated quantities which require the use of the general algorithm for evaluating the maximum flow of the network. The logical cutting tree algorithm model of the network maximum flow is built using both the logical thought of the fault tree method and cutting set theories of the graphics. Therefore, the problem is resolved well and its calculation process is simpler and its logic is better than the original algorithm method.

A Theory of Complexity for Continuous Time Systems

We present a model of computation with ordinary differential equations (ODEs) which converge to attractors that are interpreted as the output of a computation. We introduce a measure of complexity for exponentially convergent ODEs, enabling an algorithmic analysis of continuous time flows and their comparison with discrete algorithms. We define polynomial and logarithmic continuous time complexity classes and show that an ODE which solves the maximum network flow problem has polynomial time complexity. We also analyze a simple flow that solves the Maximum problem in logarithmic time. We conjecture that a subclass of the continuous P is equivalent to the classical P.

TWO EFFICIENT ALGORITHMS FOR THE GENERALIZED MAXIMUM BALANCED FLOW PROBLEM

Minoux considered the maximum balanced flow problem, i.e. the problem of finding a maximum flow in a two-terminal network N=(V, A) with source s and sink t satisfying the constraint that any arc-flow of N is bounded by a fixed proportion of the total flow value from s to t, where V is vertex set and A is arc set. Several efficient algorithms, so far, have been proposed for this problem. As a natural generalization of this problem we focus on the problem of maximizing the total flow value of a generalized flow in a network N=(V, A) with gains γ(a)>0 (a∈A) satisfying any arc-flow of N is bounded by a fixed proportion of the total flow value from s to t, where γ(a)f(a) units arrive at the vertex w for each arc-flow f(a) (a≡(v, w)∈A) entering vertex v in a generalized flow in AF. We call it the generalized maximum balanced flow problem and if γ(a)=1 for any a∈A then it is a maximum balanced flow problem. The authors believe that no algorithms have been shown for this generalized version. Our main results are to propose two polynomial algorithms for solving the generalized maximum balanced flow problem. The first algorithm runs in O(mM(n, m, B')log B) time, where B is the maximum absolute value among integral values used by an instance of the problem, and M(n, m, B') denotes the complexity of solving a generalized maximum flow problem in a network with n vertices, and m arcs, and a rational instance expressed with integers between 1 and B'. In the second algorithm we combine a parameterized technique of Megiddo with one of algorithms for the generalized maximum flow problem, and show that it runs in O({M(n, m, B')}^2)time.

On multiroute maximum flows in networks

AbstractLet G = (N, A) be a network with a designated source node s, a designated sink node t, and a finite integral capacity uij on each arc (i, j) ∈ A. An elementary K‐flow is a flow of K units from s to t such that the flow on each arcis 0 or 1. A K‐route flow is a flow from s to t that may be expressed as a nonnegative linear sum of elementary K‐flows. In this paper, we show how to determine a maximum K‐route flow as a sequence of O(min {log (nU), K}( maximum‐flow problems. This improves upon the algorithm by Kishimoto, which solves this problem as a sequence of K maximum‐flow problems. In addition, we have simplified and extended some of the basic theory. We also discuss the application of our technique to Birkhoff's theorem and a scheduling problem. © 2001 John Wiley & Sons, Inc.

On a realization problem from non‐terminal capacity matrices on undirected flow networks

AbstractThe problem of realizing a given matrix on an undirected flow network has been studied and various results have been obtained. Most of these results consist of necessary and sufficient conditions for methods of realization in which the maximum good flow between the two points and the matrix components coincide. However, there are cases in which coincidence is not required as long as they are close to each other. In this paper, we consider the realization under certain conditions when values not necessarily realizable as maximum good flows of the flow network are given between two points. First, the upper and lower bounds of the maximum good flow are given. The problem of realization between these two extremes is studied and the necessary and sufficient condition is given. Next, by means of this result, we consider the problem of minimizing the difference between the given value and the maximum good flow in the undirected flow network. A realization method is studied. © 2001 Scripta Technica, Electron Comm Jpn Pt 3, 84(12): 28–39, 2001

CALCULATING MAXIMUM ENTROPY FLOWS IN MULTI-SOURCE, MULTI-DEMAND NETWORKS

Previous work has shown how maximum entropy flows in a water distribution network can be calculated by maximizing a nodal entropy function. This requires the use of numerical nonlinear optimization. In the case of single-source networks a much simpler path entropy formulation exists which permits solutions to be obtained quickly by manual calculations not requiring numerical optimization. This paper extends the path entropy formulation to general multi-source, multi-demand networks and develops a simple manual calculation method for maximum entropy flows in these general networks. The method is quick, non-iterative and does not directly involve the use of numerical optimization. Examples are presented and discussed.

Flows in Undirected Unit Capacity Networks

We describe an O(min(m,n3/2 )m1/2 )-time algorithm for finding maximum flows in undirected networks with unit capacities and no parallel edges. This improves upon the previous bound of Karzanov and Even and Tarjan when $m = \omega(n^{3/2})$, and upon a randomized bound of Karger when $v = \Omega(n^{7/4}/m^{1/2})$.

On the time complexity of minimum and maximum global snapshot problems

Deriving the minimum and maximum global snapshots is very useful for some error detection problems in distributed programs. Several researchers, e.g., Groselj, Chen and Wu, have shown that the minimum and maximum global snapshot problems are linear-time reducible to the maximum constant-ratio network flow (MCNF) problem, here defined as the wellknown maximum network flow problem with m = Θ( n), where m is the number of edges and n is the number of vertices in the given flow network. In this paper we show in a reverse way that the MCNF problem is also linear-time reducible to these global snapshot problems. Thus, we can conclude that the global snapshot problems are “as difficult as” the MCNF problem in terms of time complexity.