Network Flow of Graph Theory and Its Application

Ruiheng Zhang

East China University of Science and Technology, Shanghai, China

Keywords: Graph Theory, Network Flow Algorithm, Application, Challenge.

Abstract: Network flow in graph theory is a theory that studies the most effective way to transport data or goods in

various networks. This paper comprehensively reviews the basic concepts, theorems, algorithms, and their

practical applications of network flow theory in graph theory. First, it introduces the basic concepts, origins,

development process, and application scenarios of network flow in daily life, clarifying the motivation and

purpose of studying network flow. Then, it defines some important terms and explains their relationships.

Subsequently, this paper further discusses the core theorems and algorithms of network flow, analyzing the

working principles, advantages and disadvantages, and application scenarios of each algorithm. In addition,

this paper also explores advanced issues and special applications related to network flow. In the application

and expansion section, it continues to analyze practical application cases of network flow in logistics and

supply chain optimization, communication network design, project management key path analysis, and other

fields. Finally, it discusses the current main challenges in the field of network flow and looks forward to

possible research directions in the future.

1 INTRODUCTION

Graph theory is an emerging discipline, belonging to

a branch of mathematics, it originated from the

Konigsberg seven bridge problem, since it was

proposed, has attracted the attention of many

researchers and is widely used in various disciplines,

such as computer science, electronics, information

theory, network theory, physics, operations research,

chemistry, etc. The graph is composed of several

given points and the line connecting two points, the

points in the graph represent elements, the line

connecting two points represents the specific

relationship between the two elements. In the graph

theory, the graph is the research object, usually used

to describe a certain specific relationship between

some elements. The minimum spanning tree of

connected graph, network flow problem, shortest path

problem and graph matching problem are important

problems in the study of graph theory(Wang

Rui,2014). This paper will focus on the network flow

problem.

Network flow is a very important field in graph

theory, which mainly studies how to effectively

transport substances or information between different

https://orcid.org/0009-0006-8350-7617

vertices in a directed graph. The concept of network

flow was first proposed in the 1950s, when it was

mainly used to solve transportation problems, i.e.,

how to most efficiently transport materials from one

place to another. And the relevant mathematical

model of this problem was established by L.R. Ford,

Jr. and D.R. Fulkerson. in 1956, and their research

greatly promoted the development of this field. With

the passage of time, network flow theory has not only

developed deeply in the field of mathematics but has

also gradually extended to multiple disciplines such

as computer science, engineering, and economics. It

has become a multidisciplinary research field.

The motivation for studying network flow mainly

lies in its wide application value and the demand for

many practical problems. In daily life, network flow

models can be applied to various scenarios, such as

network data transmission, water supply systems,

traffic planning, and the design and optimization of

power networks. For example, in internet technology,

network flow algorithms can help optimize the

transmission paths of data between servers to ensure

efficient and stable information transmission. In

urban traffic planning, the use of network flow

models can optimize road efficiency and reduce

congestion.

692

Zhang, R.

Network Flow of Graph Theory and Its Application.

DOI: 10.5220/0012968800004508

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 1st International Conference on Engineering Management, Information Technology and Intelligence (EMITI 2024), pages 692-696

ISBN: 978-989-758-713-9

Here is a concrete example, suppose there is a

process in which a substance passes through a system

from the source where it is produced to the sink where

it is consumed. The origin produces the substance at

a fixed rate, while the sink consumes the substance at

the same rate. Intuitively, the "flow" of matter at any

point in the system is the speed at which that matter

travels through the system. A "network flow" can be

used to model "flow" problems such as liquids

flowing through pipes, components passing through

assembly lines, currents in the power grid, and

information transmitted through communication

networks (Zhang Xiaowei, 2009). After any

optimization situation is abstracted into a network

graph, it can be converted into a maximum flow

problem, that is, to find a flow that maximizes the

total flow(Chen Tian,2018).

In addition, the study of network flow has also

driven the development of algorithms, such as the

famous Ford-Fulkerson method and Edmonds-Karp

algorithm,Dinic algorithm. These algorithms are

effective tools for solving the maximum flow

problem (Hua bo,2012). The development and

optimization of these algorithms not only solve

practical problems, but also enrich the theory of graph

theory and algorithms.

In summary, the study of network flow not only

helps to solve specific engineering and social

problems, but also promotes the development of

related mathematics and computational technology.

For scientists and engineers, deep understanding and

application of network flow theory is an important

tool for solving resource allocation problems in

complex systems.

2 MODEL FORMULATION

A network consists of a set of vertices and directed

edges connecting these vertices. This directed graph

is usually denoted as (G(V, E)), where (V) is the set

of vertices and (E) is the set of edges(

Du Linrui,2014).

And network flow is a special type of network where

each edge (u, v) has a non-negative capacity c(u,

v),indicating the capacity of an edge is the maximum

amount of data or material that can flow through it,

denoted as c(u, v). Note that if there is no edge from

vertex (u) to vertex (v), then c(u, v) = 0. The

maximum flow rate edge(u,v) is allowed to pass

through is f(u,v),and they satisfy an inequality

f(u,v)≤c(u,v)(Zhu Hui,2018). Network flow problems

mainly involve transmitting as much "flow" as

possible from a source node to a sink node in a

directed graph,in which source means the the starting

point of the network flow, a point with an in-degree

of 0 and an outflow capacity of infinity,and sink

means the ending point of the network flow, a point

with an out-degree of 0 and an inflow capacity of

infinity(Yan Jixing,2015).

In general, each edge in a directed graph has a

capacity limit mentioned before. At the same time,

the inflow and outflow of traffic at any point in the

network (except the source and sink points) must be

equal to ensure flow conservation. The source and

sink points serve as bounding points for input and

output flows in this framework, defining the

boundaries of network flow. And the research focus

of network flow problem is to find the maximum flow

from one or more sources to one or more sinks while

satisfying the above capacity constraints, flow

conservation, and non-negative flow volume.

Solving network flow problems relies on some

core theorems and algorithms. The most commonly

used is the Maximum Flow Minimum Cut theorem

and its related algorithms, such as the Ford-Fulkerson

method, Edmonds-Karp algorithm, and Dinic

algorithm. The Maximum Flow Minimum Cut

theorem is the cornerstone of network flow theory. It

points out that in a flow network, the maximum flow

from the source to the sink is equal to the minimum

capacity of any cut that separates the source and the

sink in the network(Lu Shengtao,2015). This theorem

not only provides an elegant optimal condition but

also is the theoretical basis for many algorithm

designs.

The Ford-Fulkerson method is a method to solve

the maximum flow problem, which relies on the

concept of augmenting path, which refers to a path

from the source to the sink where additional flow can

be added. In each iteration, the algorithm searches for

an augmenting path and increases the flow based on

the minimum capacity edge on this path until there

are no more augmenting paths(Liu Yaonian et

al.,2012). The time complexity of this algorithm

depends on the edge capacities in the network and

could be non-polynomial in the worst case.

The Edmonds-Karp algorithm is an improved

version of the Ford-Fulkerson method that uses

breadth-first search (BFS) to find augmenting paths,

ensuring that the shortest augmenting path is always

found(Xu Yongda,2015). This method has a time

complexity of O(V*E^2) in the worst case, where V

is the number of vertices and E is the number of

edges. The Edmonds-Karp algorithm has excellent

convergence speed, stable time performance, and is

easy to implement, but it also has limitations in

efficiency and may not perform well on some extreme

or demanding problems.

The Dinic algorithm is another efficient maximum

Network Flow of Graph Theory and Its Application

693

flow algorithm that utilizes the concept of

hierarchical graph. In each iteration, the algorithm

first builds a hierarchical graph and then searches for

multiple disjoint augmenting paths in the hierarchical

graph. This layered approach reduces the need for

repeated searches and lowers the time complexity of

the algorithm to O(V^2 * E). The Dinic algorithm

performs particularly well on sparse graphs,

especially when the graph scale is large.

In general, each of these algorithms has its own

advantages and disadvantages. The flexibility of the

Ford-Fulkerson method makes it intuitive to

understand and implement, but it may not be efficient

in some cases. The Edmonds-Karp algorithm

improves efficiency through a fixed search strategy

and is suitable for network flow problems with

smaller edge capacities or denser graphs. The Dinic

algorithm, with its more sophisticated hierarchical

strategy, provides better performance in many

practical applications and is particularly suitable for

handling large network flow problems. When

choosing a specific algorithm, it is necessary to

balance the scale and characteristics of the actual

problem.

3 EXTENSION WORK

The basic model of network flow problems can be

extended to more complex scenarios and special

graph structures to handle a wider range of practical

applications. Next,several advanced issues in

network flow will be explored, including multi-

commodity flow problems, minimum cost flow

problems, and the applications of these problems in

specific types of graphs, such as planar graphs and

directed acyclic graphs (DAGs).

Multi-commodity flow problems are an extension

of network flow theory, where multiple different

commodities (or flows) need to be transported in the

same network. Each commodity starts at a specific

source and is transported to a specific sink. The

challenge of this problem is that the flows of different

commodities may have conflicts on the paths in the

network, i.e., multiple flows need to share the

capacities of the same edges. Therefore, the key is

how to allocate the capacity of each edge in the

network to maximize the total flow of various

commodities without violating the edge capacity

limits. Solutions usually involve complex

optimization algorithms such as linear programming.

Minimum cost flow problems are another form of

network flow problem that not only considers the

maximization of flow but also the minimization of

cost. In this problem, each edge not only has capacity

limits but also has a unit flow transmission cost. The

goal is to find a configuration of flow that minimizes

the total cost from the source to the sink while

satisfying certain flow requirements. This type of

problem is widely used in cost-sensitive

transportation and distribution systems, such as

supply chain management and power networks.

The application of network flow in special

attribute graphs is mainly as follows:

Planar graph is a special type of graph that can be

drawn on a plane without crossing edges. The

application of network flow problems in planar

graphs is particularly interesting because the structure

of planar graphs allows certain network flow

algorithms (such as shortest path finding) to be

executed more efficiently. For example, the

bidirectionality of planar graphs can be used to

quickly determine the direction and magnitude of

flow, thereby optimizing algorithm performance.

While in directed acyclic graphs,vertices can be

naturally sorted so that all edges point from vertices

with smaller sequence numbers to those with larger

ones. This structure enables network flow problems

to be solved more efficiently through dynamic

programming and other methods. For example,

starting from the source point, you can gradually

move forward and calculate the maximum flow to

each vertex until reaching the sink point. The nature

of DAG simplifies the calculation process, reduces

circular dependency problems, and is particularly

suitable for dealing with problems such as project

scheduling and resource allocation.

These advanced network flow problems and

applications of special graph structures demonstrate

the flexibility and powerful functionality of network

flow theory, which can adapt to diverse practical

needs and complex environments. These theories not

only have wide applications in mathematics and

computer science, but also have important practical

significance in engineering, economics, social

sciences and other fields.

Network flow theory has demonstrated its unique

value and wide applicability in many practical

applications. Through accurately simulating the flow

of resources in the network, this theory not only

optimizes various industries but also promotes the

development and implementation of new

technologies. This section will detail the application

of network flow in areas such as logistics and supply

chain optimization, communication network design,

and key path analysis in project management.

Firstly, logistics and supply chain management

are one of the most widely applied fields of network

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694

flow theory. In this field, network flow models can

effectively optimize the distribution of goods and

services and minimize transportation costs and time.

By building a network model that includes multiple

supply points, warehouses, and sales points, and

utilizing algorithms such as shortest path algorithms

or maximum flow-minimum cut theorems,

companies can design the most cost-effective and

efficient supply chain network structure. For

example, international retail giant Wal-Mart uses

network flow models to optimize its global supply

chain, significantly improving supply efficiency and

reducing operating costs.

Secondly, communication network design also

relies on network flow theory to enhance service

quality and coverage. In communication networks,

the transmission of data packets can be viewed as a

kind of flow, and network flow algorithms are used to

ensure that data is transmitted most efficiently from

the source to the destination within maximum

capacity limits. In addition, network flow models are

also used to optimize data routing and bandwidth

allocation in networks to prevent network congestion

and improve data transmission stability. For example,

Google applies customized network flow algorithms

in the design of its data center networks to optimize

data flow worldwide.

Finally, the Critical Path Analysis (CPA)

technique in project management is another example

of the application of network flow theory. By

establishing a network model of project activities and

time, CPA helps project managers identify the most

critical sequence of activities in terms of time, which

means any delay will directly affect the project's

completion date. The application of this method

significantly improves the efficiency and

effectiveness of managing complex projects, and it is

widely used in construction and engineering projects

to ensure the completion of projects on schedule.

In summary, the application of network flow

theory is not limited to optimizing the performance of

existing systems. Its potential in emerging

technologies and complex system analysis remains to

be further explored. Future research can explore its

application possibilities in more fields, such as smart

city traffic systems, energy distribution networks, etc.

4 CONCLUSIONS

Network flow problems, as an important field in

operations research and computer science, have been

widely used in many practical problems. However,

with the increasing application requirements and

technological development, the field of network flow

still faces many challenges, which require continuous

theoretical and technological innovation to cope with.

Firstly, Many network flow algorithms are

theoretically effective, but their complexity is high in

complex network structures or large data scenarios,

making it difficult to achieve fast and effective

solutions. Also,With the sharp increase in data

volume and network scale, existing algorithms face

efficiency and storage pressure when dealing with

large-scale network flow problems. In addition,real-

world networks are often dynamically changing, such

as traffic flow and information flow. Existing static

network flow models are difficult to effectively

capture and handle such dynamics. Finally,real-world

network flow problems often involve the trade-off

between multiple objectives, such as cost, time, and

reliability. It is necessary to develop complex

network flow algorithms that can handle multi-

objective optimization.

Therefore,it is important to research more

efficient algorithms or improve existing algorithms to

cope with large-scale and dynamic network flow

problems in the future. For example, using machine

learning methods to predict and optimize the

distribution and changes of network flows.

Besides,integrated network models can be built which

will simultaneously handle multiple types of flows

(such as data flow, material flow, financial flow, etc.)

to comprehensively solve complex practical

problems.

With technological advancements, emerging

fields such as the Internet of Things and smart cities

provide new application scenarios for network flow

problems. It is necessary to study network flow

theories and technologies that are suitable for these

scenarios. So combining theories and methods from

other disciplines such as economics and sociology to

develop new network flow models and algorithms is

a good idea to better address the network issues.

Network flow problems, as a bridge connecting

theory with practice, are not only of great significance

to scientific development but also have a profound

impact on social and economic activities. Through

continuous theoretical innovation and technological

development, the field of network flows is expected

to solve more complex practical problems and

achieve broader applications in the future.

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