1. Graph Theory Modeling – A Petri Net based Approach
Shi-Jian Liu, Xing-Si Xue, and Jing Zhang

2. Outline Introduction Problem Statement and Preliminaries
Modeling of Maximum Flow using Petri nets
Conclusion and Future Works
Q & A

3. Introduction Outline Graph & Petri Net theory Motivation Contributions
Problem Statement and Preliminaries
Modeling of Maximum Flow using Petri nets
Conclusion and Future Works
Q & A

4. Graph theory and problems
G = (V, E). V: Vertexes, E: Edges
Shortest path : Dijkstra, A*
Traveling salesman problem : NP-hard
Maximum flow/Minimum cut : Ford-Fulkerson method
There are many classic problems in graph theory, including the shortest path
problem, maximum flow problem, etc. Many algorithms have been proposed to
solve these problems, but it is undeniable that both the theories and solutions of
the graph theory problems are generally based on abstract concepts and difficult
to understand especially for junior students.
旅行商（salesperson)问 题 ，要求找出经过一个加权图中所有顶点的最短路径长度。

5. Petri Nets (PNs)
A Petri net also known as a place/transition net which consists of elements such as, the Place (Circle), Transition (Rectangle) and Token (Dot).
graphical notation for stepwise processes that include choice, iteration, and concurrent execution,
the most used formalism for modeling and analyzing.
Dining philosophers problem

6. Motivation Challenge:
Graph theory problems have played fundamental and crucial roles in many researches
Road networks evaluation [1]
Online semi-supervised learning [2]
Energy minimization in vision : Graph Cuts Framework [3]
The descriptions and solutions are usually abstract and difficult for understanding and expressing
[1] V. Ramesh, S. Nagarajan, J. J. Jung, S. Mukherjee, Max-ow min-cut algorithm with application to road networks, Concurr. Comp.-Pract. E 29 (11) (2017) e4099.
[2] L. Zhu, S. Pang, A. Sarrafzadeh, T. Ban, D. Inoue, Incremental and decremental max-ow for online semi-supervised learning, IEEE T. Knowl. Data En. 28 (8) (2016)
[3] Y. Boykov, O. Veksler, R. Zabih, Fast approximate energy minimization via graph cuts, IEEE T. Pattern. Anal. 23 (11) (2001)

7. Motivation Solution Aim Adopt PNs for modeling and communication
use PNs to formalize the max-flow/min-cut problem.
visualize the classic Ford-Fulkerson algorithm.

8. Contributions
PNs models for flow networks and residual networks are proposed in this paper, which are intuitive to understand.
Simulation methods are presented to solve and illustrate the max-flow/min-cut of a given flow network.

9. Problem Statement and Preliminaries
Outline
Introduction
Problem Statement and Preliminaries
Flow networks and flows
Petri nets
Modeling of Maximum Flow using Petri nets
Conclusion and Future Works
Q & A

10. Flow networks Definition 1 (Flow networks [1]). Denition 2 (Flow [1])
Directed graph G = (V;E)
Nonnegative capacity C(u; v)
No self-loops
A source s and a sink t …
Denition 2 (Flow [1])
A real-valued function f : V×V  R 2
[1] T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms, Third Edition, 3rd Edition, The MIT Press, 2009.

11. Petri Net A triplet N = (S, T; F) satisfying
where
dom means domain of F and meets
cod means codomain of F and meets
S and T denote the set of Place and Transition.
F denotes the relation between S and T.

12. Outline Modeling of Maximum Flow using Petri nets
Introduction
Problem Statement and Preliminaries
Modeling of Maximum Flow using Petri nets
Ford-Fulkerson method
Flow network modeling
Residual network modeling
PNs based Max-flow/min-cut Simulation
Conclusion and Future Works
Q & A

13. Ford-Fulkerson method

14. Flow network modeling

15. Residual network modeling

16. PNs based Max-flow/min-cut Simulation
Given a flow network, we can construct a PN system accordingly.
A maximum flow is achieved when there exists no path for a token transmitted from source Place to sink Place.
The entire procedures can be intuitive demonstrated by automatically executed simulation.
VisObjNet

17. Conclusion and Future Work Q & A
Outline
Introduction
Problem Statement and Preliminaries
Modeling of Maximum Flow using Petri nets
Conclusion and Future Work
Q & A

18. Conclusion and Future Work
Benefit from animated graphical notations of PNs, a PNs based maximum flow modeling approach is introduced in this paper.
Modeling methods for both flow networks and residual networks are firstly proposed.
Based on the proposed models, a PN system for solving the maximum flow is presented.
In the further
Validate the proposed model by proving that it is equivalent to the Ford-Fulkerson method.
It is feasible for us to extent this work to other problems in graph theory.

19. Thank you for your attention!
Q. & A.
Thank you for your attention!
Corresponding author: Shi-Jian Liu Ph.D.,
Homepage:
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Affiliation: School of Information Science and Engineering, 
Fujian Provincial Key Laboratory of Big Data Mining and Applications,
Fujian University of Technology
Address: No3 Xueyuan Road, University Town, Minhou, Fuzhou City, Fujian Province, China