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10KHEEPEE30nACH0CTb
Peeuon u Mup, 2023, № 2
Enhancing Cyber Security with Maximum Coupling in Bipartite Graphs
Sargsyan Garegin V
Candidate of Sciences in Physics and Mathematics Yerevan State University (Yerevan, RA) garegin.sargsyan.v@gmail.com Matevosyan Artur V. Master of Electronic Design Automation (EDA), 2nd year National Polytechnic University of Armenia (Yerevan, RA) garegin.sargsyan.v@gmail.com, artur.matevosyan99@gmail.com
Avetyan David R.
Master of Electronic Design Automation (EDA), 2nd year National Polytechnic University of Armenia (Yerevan, RA)
avidavetyan15@gmail.com
UDC: 519.174.7; EDN: JKCZSU; DOI: 10.58587/18292437-2023.2-95
Keywords: Network security, Edge, Bipartite graph, C++, Matrix, Security policy, Maximum coupling, Net, QT, signal-slot
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Повышение кибербезопасности с максимальной связью в двудольных графах
Саргсян Гарегин В.
Кандидат физико-математических наук Ереванский государственный университет (Ереван, РА)
garegin.sargsyan.v@gmail.com Матевосян Артур В.
Магистр автоматизации электронного проектирования (EDA), 2 курс Национальный политехнический университет Армении (Ереван, РА)
artur.matevosyan99@gmail.com Аветян Давид Р.
Магистр автоматизации электронного проектирования (EDA), 2 курс Национальный политехнический университет Армении (Ереван, РА)
davidavetyan 15@gmail.com
Аннатация. Проблема максимальной связи в двудольных взвешенных графах имеет важные практические приложения в области обороны и безопасности. Например, в контексте пограничной безопасности проблема возникает при распределении ресурсов для мониторинга и реагирования на потенциальные угрозы, где веса на краях представляют эффективность различных ресурсов для обнаружения или смягчения конкретных угроз. Нахождение максимальной связи в таком графе может гарантировать, что наиболее эффективные ресурсы будут выделены наиболее важным областям, тем самым улучшив безопасность границ.
Помимо военных приложений, проблема максимальной связи в двудольных взвешенных графах также может повысить безопасность сети. Путем моделирования сети в виде двудольного графа с узлами, представляющими компоненты сети, и ребрами, представляющими их соединения, можно использовать метод поиска максимальной связи для выявления потенциальных нарушений безопасности. Любой несопоставленный узел на графике предполагает, что компонент может быть скомпрометирован или уязвим для атаки, что делает метод поиска максимальной связи эффективным инструментом для определения приоритетов и повышения безопасности сети.
В этой статье представлен новый метод эффективного нахождения максимальной связи в двудольном взвешенном графе. В предлагаемом методе используется итеративный подход, основанный на традиционном алгоритме увеличивающего пути, но включающий дополнительные эвристики и методы сокращения для уменьшения вычислительной сложности. Экспериментальные результаты показывают, что предложенный метод превосходит существующие алгоритмы как по точности, так и по скорости. Эффективность и действенность метода делают его ценным инструментом для усиления защиты и безопасности в различных приложениях, включая военные операции и сетевую безопасность.
Ключевые слова: сетевая безопасность, ребро, двусторонний граф, c++, матрица, политика безопасности, максимальная связь, сеть, QT, сигнал-слот
Introduction
Develop and implement a software tool that, upon inputting orthogonal bipartite graphs with the corresponding file description or a graphical interface, will construct their corresponding bipartite graphs, find the maximum matching of the graph, and represent them in the software's graphical environment. The Hungarian (Munkres) algorithm [1] is used to find the maximum matching in the bipartite graph.
Let iit^ii K,-'") be the given bipartite graph. Consider the steps of the algorithm:
1. The bipartite graph corresponding matrix A is constructed with the following principle: the number of rows is equal to m = | tfj |, the number of columns is =
¿Lj = F = 1,, mj = 1,, ft, where / is the
graph's matching function, and (FyWjJ the edge
connecting vertices If these
vertices are not adjacent, then = ft.
2. If the obtained matrix is not square, i.e., m^n, then zero rows (columns) are added to the matrix to make it square.
3. As the algorithm searches for the maximum matching, all elements of the matrix are multiplied by -1, and their sum is added to the largest value of the initial matrix elements.
4. For each row, the smallest element is found and subtracted from all the elements of that row.
5. For each column, the smallest element is found and subtracted from all the elements of that column.
6. The minimal number of horizontal (row) lines is determined to cover all the zeros in the matrix. If this number is less than n, the optimal coverage has not been found, and the algorithm
continues with the next step. If this number is equal to n, the optimal coverage has been found, and it is determined according to the rows where the zeros are placed without duplicating the columns.
Figure 1: Bipartite graph
7. We are looking for the smallest uncovered element from the matrix, which is subtracted from the uncovered elements, and added to those elements that are located at the intersection points of the horizontal and vertical lines composing the matrix cover. Proceeding to the 6th step of the algorithm. The time complexity of the algorithm is t [3]: Let's consider an example of the algorithm's operation on the following bipartite graph:
For graph in Fig
[a.cre^ = [brdrfi
1. The matrix will have the following form:
1,
2.
/2 £ 5 \
[ 20 19 I
\ 0 ft /
The matrix is square, so no action will be performed at this step.
/2 5 a \ [ 20 19 J
I 0 0/
3. After performing the actions of the 3rd step, we get the following matrix:
/3 a
[20 0 SI I
UQ le 40/
4. Subtracting the smallest element from each
row, we get:
/ 3 0 5 \ [20 0 21J
e ao/
Subtracting the smallest element from each column, we get:
5.
/ О 0 Q\ f 17 0 161 427 0 25/
6. To cover this matrix, 2 lines are needed, which is less than 3, so we proceed to the next step of the algorithm:
7.
С
Регион и мир, 2023, № 2
0 0 0\ 17 О 16
27 О 2В/
The smallest uncovered element is 16, so as a result of the 7th step, we get the following matrix:
0 16 Ф 10 0 11 О
Repeating the 6th step of the algorithm, we find that the zeros are covered in 3 lines, so the optimal cover is found
0 16 Г 10 0 11 0 w
According to the lines and columns covering zeros, the following will be:
0 16 Ф 10 0 11 0 9J
In the initial matrix:
20 40 19 30
As a result, we find that the maximum matching will be the following: {(a,b),(c,f),(e,d)}.
J w
с
.ix:
I2 20
V 0
Software Implementation
During the execution of the work, a graphical user interface (GUI) for mapping orthogonal linesegment [2] domains has been developed, and the algorithm for finding the maximum matching in a bipartite graph generated from the input domains has been implemented. The main components of the software are:
• Implementation of the algorithm
• Graphical interface for displaying orthogonal domains and the corresponding partitions
• Graphical component for representing the partition graph and its maximum matching
During the operation of the software, orthogonal domains are read from an input file or entered by the user, which are stored in a specific data structure and graphically represented. Afterward, when the user presses the "show graph" button, the graph corresponding to the domains is created and its maximum matching is found.
The graphical interface receives the data structure, which describes the bipartite graph and the maximum matching, which are displayed in the graphical environment.
Description of the input file
The file describing the orthogonal directionality domains can have any extension. Below, let's examine an example of an input file describing the domains:
Figure 2: Example of input file 3 testtxt- Notepad File Edit Format View Help
S3 LayoutMatching - □ X
The part for displaying the orthogonal directionality domains consists of the following sections:
• LayoutMatching - a main window inherited from QmainWindow that represents a template containing the basic elements of the graphical interface (QWidget, QPushButton, QStatusBar, etc.). Using Qt's signal-slot mechanism [5], it connects the different parts of the interface, specifically when the "Calculate Matching" button is pressed to calculate the maximum matching, this class's function is called, which, with the help of another part of the program, obtains the graph constructed from the orthogonal directionality domains, passes it to the algorithm, and, as a result, obtaining the maximum matching, opens the window displaying the graph, transferring the information about the graph.
• CanvasWidget - A class inherited from QWidget that is responsible for drawing rectangular areas and their divisions. The base class QWidget's paintEvent(), mousePressEvent(), mouseMoveEvent(), and mouseDoubleClickEvent() functions are overridden to allow the user to add new areas during the program's operation. To add a new area, the user clicks once at the point where the rectangular area begins, then moves the mouse to draw the side, depending on the direction -horizontal or vertical, and then clicks again to start drawing the next side. To finish, the user doubleclicks at the endpoint of the last side, which closes the boundary and constructs the area.
Figure 4: The view of the maximum matching of the
_graph_
E Graph viewer — □ X
The window representing the graph, by receiving the graph and a certain number of its edges as input, depicts the graph's vertices in a circular layout, while the edges belonging to the
{
{(20,20)(20j50)(40,50)} {(20,20) (20 j 50) (4-0,50) (4-0,20) } {(100,25)(100,45)(150,45)(150,25)} {(120,30)(120,60)(140,60)(140,30)}
{(240,240)(240,250)(270,250)(270,240)} }
{
{(200,200)(200,300)(300,300)(300,200)} {(210,210)(210,290)(290,290)(290,210)} {(210,210)(210,280)(280,280)(280,210)}
{(210,210)(210,220)(220,220)} }
{
{(20,20)(20,50)(40,50)(40,20)}
{(25,25)(25,45)(35,45)(35,25)} }
{
{(10*25)(10,45)(35,45)(35,25)} }
{
{(27,30)(27,40)(32,40)(32,30)} }
{
{(4^,25)(45,45)(65,45)(65,25)} }
Structure of the graphical interface The graphical interface consists of a main part, where the orthogonal directionality domains and their divisions are depicted, and a button that launches the main algorithm [4]. Figure 3 presents the appearance of the graphical environment.
The user can load the orthogonal directionality domains described in the input file using the File->Open button. When the user presses the "Calculate Matching" button, a separate window opens, displaying the graph corresponding to the domains and the maximum matching in that graph.
_Figure 3: Main part of the graphical interface_
input set of edges are highlighted with a different color. In this window, the user has the ability to change the appearance of the vertices, their sizes, as well as change the arrangement of the vertices relative to each other.
Obtained results of the software tool
By providing the file shown in Fig. 2 as input data to the software tool, the result of the work is the graph depicted in Fig. 4, where the edges forming the maximum matching are shown in a distinct color.
Conclusions
In this work, we have developed an algorithm for finding the maximum matching in a bipartite graph with bent edges, using an orthogonal directionality language to describe areas. Our implementation includes a software tool that graphically displays the input areas and calculates the maximum matching for the corresponding bipartite graph. The tool presents the resulting graph in a graphical environment, with the edges forming the maximum matching displayed in a different color. The software implementation materials are available at the following link: [6].
The algorithm can be used in network security to identify potential vulnerabilities or breaches. By modeling a network as a bipartite graph and searching for the maximum matching, any unmatched nodes can indicate potential points of weakness. This makes the maximum matching search algorithm a valuable tool for prioritizing and
Регион и мир, 2023, № 2
improving network security.
Overall, our algorithm and software tool offer a flexible and efficient approach to finding the maximum matching in bipartite graphs, with important practical applications in defense and security. The availability of our software implementation materials at the given link will enable others to build on our work and apply the algorithm in a variety of contexts.
References
1. Stroustrup - The C++ Programming Language 4th edition, - Addison-Wesley Professional, 2012. -1040 p.
2. Mark Allen -Data structures and algorithm analysis in C++, 2006, 507p
3. Kuhn H. W., 50 Years of Integer Programming 1958-2008. Springer, Berlin, Heidelberg, 2010, p29-47
4. Petrosyan P. A, Mkrtchyan V. V., Kamalyan R. R,
Graph Theory, YSU Publishing House, 2015.
5. Jasmin Blanchette, Mark Summerfield -C++
GUI programming with Qt 4 -Prentice Hall PTR, 2006. - 560 p.
6. https://github.com/davidavetyan/LayoutMatching
Сдана/^шйШф11 03.03.2023 Рецензирована/0-рш^пиф1 f 11.04.2023 Принята/^йцтй^Щ f 17.04.2023
