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

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

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

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。

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

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) 2115-2127. [3] Y. Boykov, O. Veksler, R. Zabih, Fast approximate energy minimization via graph cuts, IEEE T. Pattern. Anal. 23 (11) (2001) 1222-1239.

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.

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.

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

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.

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.

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

Ford-Fulkerson method

Flow network modeling

Residual network modeling

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

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

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.

Thank you for your attention! Q. & A. Thank you for your attention! Corresponding author: Shi-Jian Liu Ph.D., Email: liusj2003@126.com  Homepage: http://bigdata.fjut.edu.cn/sjliu/  Tel.: +86-18359167583 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