Maximum Flow In Network Research Articles

The problem of finding the maximum multiple flow in the divisible network and its special cases

In the article the problem of finding the maximum multiple flow in the network of any natural multiplicity k is studied. There are arcs of three types: ordinary arcs, multiple arcs and multi-arcs. Each multiple and multi-arc is a union of k linked arcs, which are adjusted with each other. The network constructing rules are described. The definitions of a divisible network and some associated subjects are stated. The important property of the divisible network is that every divisible network can be partitioned into k parts, which are adjusted on the linked arcs of each multiple and multi-arc. Each part is the ordinary transportation network. The main results of the article are the following subclasses of the problem of finding the maximum multiple flow in the divisible network. 1. The divisible networks with the multi-arc constraints. Assume that only one vertex is the ending vertex for a multi-arc in s network parts. In this case the problem can be solved in a polynomial time. 2. The divisible networks with the weak multi-arc constraints. Assume that only one vertex is the ending vertex for a multi-arc in k-1 network parts (1 ≤ s < k − 1) and other parts have at least two such vertices. In that case the multiplicity of the maximum multiple flow problem can be decreased to k - s. 3. The divisible network of the parallel structure. Assume that the divisible network component, which consists of all multiple arcs, can be partitioned into subcomponents, each of them containing exactly one vertex-beginning of a multi-arc. Suppose that intersection of each pair of subcomponents is the only vertex-network source x0. If k=2, the maximum flow problem can be solved in a polynomial time. If k ≥ 3, the problem is NP-complete. The algorithms for each polynomial subclass are suggested. Also, the multiplicity decreasing algorithm for the divisible network with weak multi-arc constraints is formulated.

On the Optimal Linear Network Coding Design for Information Theoretically Secure Unicast Streaming

The continuous growth of media-rich content calls for more efficient and secure methods for content delivery. In this paper, we will address the optimal linear network coding (LNC) design for secure unicast streaming against passive attacks, under the requirement of information theoretical security . The objectives include 1) satisfying the information theoretical security requirement, 2) maximizing the transmission rate of a unicast stream, 3) minimizing the number of additional random symbols, and 4) minimizing the total bandwidth cost of content delivery. To fulfill the first three objectives, we formulate an information theoretically secure unicast streaming (ITSUS) problem, and then solve it by transforming it to a maximum network flow problem with node-capacity constraints. Based on the solution of the ITSUS problem, we develop an efficient algorithm that can find the optimal transmission topology with minimum bandwidth cost in a polynomial amount of time. With the optimal transmission topology, we investigate the design of both deterministic LNC and random LNC. For the deterministic LNC design, we not only prove that it achieves the four objectives but also analyze the size of required finite field. Moreover, for the random LNC design, we analyze the probability that a random LNC scheme satisfies the information theoretical security requirement. Finally, extensive simulation experiments have been conducted, and the results demonstrate the effectiveness of the proposed algorithms.

Orthogonal graph drawing with inflexible edges

We consider the problem of creating plane orthogonal drawings of 4-planar graphs (planar graphs with maximum degree 4) with constraints on the number of bends per edge. More precisely, we have a flexibility function assigning to each edge e a natural number flex(e), its flexibility. The problem FlexDraw asks whether there exists an orthogonal drawing such that each edge e has at most flex(e) bends. It is known that FlexDraw is NP-hard if flex(e)=0 for every edge e[1]. On the other hand, FlexDraw can be solved efficiently if flex(e)≥1[2] and is trivial if flex(e)≥2[3] for every edge e.To close the gap between the NP-hardness for flex(e)=0 and the efficient algorithm for flex(e)≥1, we investigate the computational complexity of FlexDraw in case only few edges are inflexible (i.e., have flexibility 0). We show that for any ε>0FlexDraw is NP-complete for instances with O(nε) inflexible edges with pairwise distance Ω(n1−ε) (including the case where they induce a matching), where n denotes the number of vertices in the graph. On the other hand, we give an FPT-algorithm with running time O(2k⋅n⋅Tflow(n)), where Tflow(n) is the time necessary to compute a maximum flow in a planar flow network with multiple sources and sinks, and k is the number of inflexible edges having at least one endpoint of degree 4.

The Minset-Poset Approach to Representations of Graph Connectivity

Various instances of the minimal-set poset (minset-poset for short) have been proposed in the literature, e.g., the representation of Picard and Queyranne for all st -minimum cuts of a flow network. We begin with an explanation of why this poset structure is common. We show any family of sets F that can be defined by a “labelling algorithm” (e.g., the Ford-Fulkerson labelling algorithm for maximum network flow) has an algorithm that constructs the minset poset for F . We implement this algorithm to efficiently find the nodes of the poset when F is the family of minimum edge cuts of an unweighted graph; we also give related algorithms to construct the entire poset for weighted graphs. The rest of the article discusses applications to edge- and vertex connectivity, both combinatorial and algorithmic, that we now describe. For digraphs, a natural interpretation of the minset poset represents all minimum edge cuts. In the special case of undirected graphs, the minset poset is proved to be a variant of the well-known cactus representation of all mincuts. We use the poset algorithms to construct the cactus representation for unweighted graphs in time O ( m +λ 2 n log (n/λ)) (λ is the edge connectivity) improving the previous bound O (λ n 2 ) for all but the densest graphs. We also construct the cactus representation for weighted graphs in time O ( nm log( n 2 / m )), the same bound as a previously known algorithm but in linear space O ( m ). The latter bound also holds for constructing the minset poset for any weighted digraph; the former bound also holds for constructing the nodes of that poset for any unweighted digraph. The poset is used in algorithms to increase the edge connectivity of a graph by adding the fewest edges possible. For directed and undirected graphs, weighted and unweighted, we achieve the time of the preceding two bounds, i.e., essentially the best-known bounds to compute the edge connectivity itself. Some constructions of minset posets for graph rigidity are also sketched. For vertex connectivity, the minset poset is proved to be a slight variant of the dominator tree. This leads to an algorithm to construct the dominator tree in time O ( m ) on a RAM. (The algorithm is included in the appendix, since other linear-time algorithms of similar simplicity have recently been presented.)

The Maximum Flows in Planar Dynamic Networks

An nontrivial extension of the maximal static flow problem is the maximal dynamic flow model, where the transit time to traverse an arc is taken into consideration. If the network parameters as capacities, arc traversal times, and so on, are constant over time, then a dynamic flow problem is said to be stationary. Research on flow in planar static network is motivated by the fact that more efficient algorithms can be developed by exploiting the planar structure of the graph. This article states and solves the maximum flow in directed (1, n) planar dynamic networks in the stationary case.

Cooperative Strategies for Maximum-Flow Problem in Uncertain Decentralized Systems Using Reliability Analysis

This study investigates a multiowner maximum-flow network problem, which suffers from risky events. Uncertain conditions effect on proper estimation and ignoring them may mislead decision makers by overestimation. A key question is how self-governing owners in the network can cooperate with each other to maintain a reliable flow. Hence, the question is answered by providing a mathematical programming model based on applying the triangular reliability function in the decentralized networks. The proposed method concentrates on multiowner networks which suffer from risky time, cost, and capacity parameters for each network’s arcs. Some cooperative game methods such asτ-value, Shapley, and core center are presented to fairly distribute extra profit of cooperation. A numerical example including sensitivity analysis and the results of comparisons are presented. Indeed, the proposed method provides more reality in decision-making for risky systems, hence leading to significant profits in terms of real cost estimation when compared with unforeseen effects.

Pattern matching with wildcards and length constraints using maximum network flow

In 2006 Chen et al. introduced an interesting text pattern matching problem with unique features. A pattern is described by a sequence of letters (literals) separated by a number of wildcards. For each pair of consecutive literals in the pattern description, a local length constraint specifies the minimum and maximum number of wildcards. Also, the pattern must occur in the text within a window for which the minimum and maximum lengths are specified by the global length constraint. Text letters are exclusively assigned to occurrences (the one-off condition). The objective is to find the maximum number of occurrences of the pattern in the text with all the constraints. For the problem (in this paper we call it Problem 1), there is a polynomial time online algorithm which does not guarantee an optimal solution. There are also polynomial time heuristic algorithms for the problem, and for some of its restricted cases. It is not known if Problem 1 is NP-hard. In this work, for two special cases of Problem 1, we give reductions to the maximum flow problem which can be solved in polynomial time.

Flow-based vulnerability measures for network component importance: Experimentation with preparedness planning

This work develops and compares several flow-based vulnerability measures to prioritize important network edges for the implementation of preparedness options. These network vulnerability measures quantify different characteristics and perspectives on enabling maximum flow, creating bottlenecks, and partitioning into cutsets, among others. The efficacy of these vulnerability measures to motivate preparedness options against experimental geographically located disruption simulations is measured. Results suggest that a weighted flow capacity rate, which accounts for both (i) the contribution of an edge to maximum network flow and (ii) the extent to which the edge is a bottleneck in the network, shows most promise across four instances of varying network sizes and densities.

ANALISIS SISTEM JARINGAN TRANSPORTASI DAN MODEL GRAVITASI DI KOTA AMBON

This study aimed to analyze the maximum capacity of roads in overcoming bottlenecks using the Maximum Network Flow Model and analyze the interaction of the distribution of the movement with the city located around the city of Ambon by using Gravity Model. Data obtained using the method of documentation. Obtained results, the maximum number of cars that can pass through the road network in 1 hour is 68 units. Line 1 is the terminal Mahardika to Kudamati has exceeded the capacity for the first hour there are 104 units of the car were gone and behind 102 cars. While the other track is still low, so that the growth and development of land transportation is strongly recommended. Results of the analysis of the gravitational interaction with the transport of the most powerful city of Ambon is Karang Panjang Town with great interaction and 6781.96088 weakest interaction is with great interaction Leahari City 5.27899.

The maximum flow problem in a network with special conditions of flow distribution

We consider the problem of finding a maximum flow in networks with strictly and nonstrictly determined flow distribution. We show that, for each condition of flow distribution, a solution to the problem under consideration exists and is unique. Some algorithms for finding a maximum flow are developed for each condition of flow distribution. We find bounds for the maximum flow value in networks with strict flow distribution.

Fast Computation of Cuts With Reduced Support by Solving Maximum Circulation Problems

We present a technique to efficiently compute optimal cuts required to solve 3-D eddy current problems by magnetic scalar potential formulations. By optimal cuts, we mean the representatives of (co)homology generators with minimum support among the ones with a prescribed boundary. In this paper, we obtain them by starting from the minimal (co)homology generators of the combinatorial two-manifold representing the interface between conducting and insulating regions. Optimal generators are useful because they reduce the fill-in of the sparse matrix and ease human-guided basis selection. In addition, provided that the mesh is refined enough to allow it, they are not self-intersecting. The proposed technique is based on a novel graph-theoretic algorithm to solve a maximum circulation network flow problem in unweighted graphs that typically runs in linear time.

Exact algorithms for solving a Euclidean maximum flow network interdiction problem

We consider an interdiction problem that involves an operator (or defender) whose goal is to maximize flow from a source node to a sink node in some network that resides in Euclidean space. The problem we examine takes the perspective of an interdictor, who seeks to minimize the defender's maximum flow by locating a set of attacks that diminish arc capacities in accordance with the distance from the arc to the attack. Attacks are not restricted to node or arc locations, and can occur anywhere on the region in which the network is located. We refer to this problem as the Euclidean maximum flow network interdiction problem (E‐MFNIP). We show that E‐MFNIP is NP‐hard, as it generalizes the maximum flow interdiction problem studied by Wood . This article contributes two approaches to solving E‐MFNIP based on solving a sequence of lower‐bounding integer programs from which upper bounds can be readily obtained, and shows that these bounds are convergent. Computations on a set of test instances indicate that an approach based on space‐discretization tends to converge much faster than one based on linearizing the nonlinear capacity functions. We demonstrate the application of our space‐discretization approach on a real geographical network. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 64(2), 109–124 2014

Capacity factors in a point-to-point network

In this study, we investigate some properties of capacity factors (CFs) that may affect the link failure problem in network coding. Roughly speaking, a CF is an edge set where its deletion will reduce the maximum flow, whereas deleting any proper subset of this CF will not. More generally, a k-CF is a minimal (not minimum) edge set, where its removal will decrease the maximum network flow by at least k. First, under a point-to-point acyclic scenario, we characterize all the edges contained in some CFs and we propose an efficient classification algorithm. We show that all of the edges on some s–t path in a point-to-point acyclic network belong to some 2-CF, and we also present some other properties of CFs. Some computational hardness results related to CFs are obtained. We prove that determining the existence of a CF in a cyclic network with a size greater than a given number is an NP-complete problem and that the time complexity of computing its capacity rank is lower bounded by that of solving its maximal flow. In addition, we propose an analogous definition of CFs on vertices as a special case that captures edge CFs.

Chance distribution of the maximum flow of uncertain random network

A network is called uncertain random network if some arc capacities of the network are uncertain variables and others are random variables. The main purpose of this paper is to study the maximum flow in an uncertain random network. Under the framework of chance theory, this paper obtains chance distribution of the maximum flow of an uncertain random network. At the same time, the expected value of maximum flow is given for an uncertain random network. A new method is derived to calculate chance distribution of the maximum flow for an uncertain random network.

Upper bound of network capacity and a static optimal packet routing strategy

In this paper, we propose a max-flow-min-cost (MFMC) problem to obtain the upper bound of network capacity and develop an efficient packet routing strategy. The MFMC problem is formulated as two linear programs (LP). The optimal solutions of the two LPs are the upper bound of network capacity (i.e., maximum network flow rate) and an optimal packet flow pattern with minimum total free flow travel time under the maximum network flow rate, respectively. We also develop a static optimal routing strategy (SORS), which proportionally deliver packets along the path according to the optimal packet flow pattern. Simulation results indicate that the proposed SORS outperforms existing routing strategies in terms of both network capacity and traffic efficiency. Furthermore, the network capacity achieved by the SORS is very close to its upper bound.

Probabilistic Analysis on Minimum &lt;i&gt;s-t&lt;/i&gt; Cut Capacity of Random Graphs with Specified Degree Distribution

The capacity (or maximum flow) of an unicast network is known to be equal to the minimum s-t cut capacity due to the max-flow min-cut theorem. If the topology of a network (or link capacities) is dynamically changing or unknown, it is not so trivial to predict statistical properties on the maximum flow of the network. In this paper, we present a probabilistic analysis for evaluating the accumulate distribution of the minimum s-t cut capacity on random graphs. The graph ensemble treated in this paper consists of weighted graphs with arbitrary specified degree distribution. The main contribution of our work is a lower bound for the accumulate distribution of the minimum s-t cut capacity. From some computer experiments, it is observed that the lower bound derived here reflects the actual statistical behavior of the minimum s-t cut capacity of random graphs with specified degrees.

Capacity reliability of water distribution systems

Hydraulic power capacity of the water distribution network (WDN) is analyzed, and energetically maximum flows in pipes and networks are determined. The concept of hydraulic power for the analysis of WDN characteristics is presented. Hydraulic power capacity characterizes the WDN capacity to meet pressure and flow demands. A capacity reliability indicator called the surplus power factor is introduced for individual transmission pipes and for distribution networks. The surplus power factor s that characterizes the reliability of the hydraulic system can be used along with other measures developed to quantify the hydraulic reliability of water networks. The coefficient of the hydraulic efficiency ηn of the network is defined. A water distribution system in service is analyzed to demonstrate the s and ηn values in the water network in service under different demand conditions. In order to calculate the s factor for WDNs, a network resistance coefficient C was determined. The coefficient C characterizes overall head losses in water pipelines and is a basis for the s factor calculation. This paper presents a theoretical approach to determine the coefficient C through matrix equations.

A Maximum Concurrent Network Flow Based Channel Assignment Algorithm for Wireless Mesh Networks

We proposed a maximum concurrent network flow based channel assignment algorithm (MCFCA) for wireless networks. In the MCFCA algorithm, we first use the maximum concurrent network flow algorithm to compute the maximum network capacity, then we assign each link a priority according to its link throughput, finally the channel is assigned to minimize the interference between neighbor nodes in the priority order. Performance simulation shows its effectiveness in channel assignment.

On-Line Verification of Relay Protection Setting Based on Transmission Contribution Degree

In this paper, a new method that is based on the transmission contribution degree is proposed, which can effictively imprve on-line verification of relay protection setting.Actually, all protection devices of a line are treated as a whole in this method. Furthermore, the network maximum flow and complex network theory are both applied in this method, which is on the view of the structural point of the grid. The importance index of protection device in a line is determined by the transmission contribution degree. As a result, on-line verification of relay protection setting is according to the descending order of the indexes. IEEE 39-bus system is used to simulate, on the whole, the simulation and analysis of the calculation results to verify the reasonableness and correctness of the method.

Network flows modeling applied to the natural gas pipeline in Brazil

The use of natural gas during the last decades has been growing, and this trend will continue until 2030, according to the Brazilian energy plan issued by Ministry of Mines and Energy (2007a). However, the capacity of transportation is one of the important elements of the supply chain in response to the distribution market. In this context, the present study aims at identifying, in a regional and integrated way, the transportation capacity of the natural gas pipeline network. Therefore, a database and a mathematical model were developed to calculate the network maximum flow in order to enable analysis of the natural gas flow in the national network. The complex natural gas flow and the forecast of increasing demand justify the study of this issue, and the development of tools to help analyze the feasibility of implanting the decennial energy plan. The mathematical model used considers a linear programming to calculate the flows in the network. Data were collected about the operating companies, existing pipelines, handling capacity, historical flow and demand forecast. The delivery capacity in the consumer markets is also analyzed by means of economic and population index compared to the 2020 target. The results show a maximum nominal capacity of 213 MM m3 per day at reception points, and 227 MM m3 per day at delivery points. Idleness was found in the networks of the Northeast and Espírito Santo area (14% and 6.1%), and lack of capacity was found in the Southeast and South networks (11.6% and 9%), compared with the economic index of consumer centers near the networks. In comparison to the current flow, 75% of idleness was found. Idleness was identified in the networks of the Northeast and Espírito Santo areas (55% and 68%), and lack of capacity was identified in the Southeast and South networks (7% and 74%), in comparison to the 2020 target. In the analysis of the 2020 target, compared to the capacity in the metropolitan areas, lack of capacity was found in the metropolitan areas of Sao Paulo, Belo Horizonte, Fortaleza and South area (68%, 5%, 59% and 74%) and idleness was found in the areas of Vitória, Recife, Rio de Janeiro, Maceió, Aracaju, Salvador and Natal (85%, 72%, 40%, 93%, 8%, 58% and 17%).