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# 数学代写|图论代写graph theory代考|Matching in General Graphs

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## 数学代写|图论代写graph theory代考|Edmonds’ Blossom Algorithm

As noted above, Jack Edmonds devised this procedure so that augmenting paths can be found within a non-bipartite graph, which can then be modified to create a larger matching [28]. To get a better understanding of the complexities that arise when searching for a matching in a non-bipartite graph, consider the two graphs $G_{8}$ and $G_{9}$ shown below, where only the graph on the left is bipartite. In both graphs, the vertex $u$ is left unsaturated by the matching shown in bold.

Notice that in $G_{8}$ above, when looking for an alternating path out of $u$, if the end of the path is in the same partite set as $u$, then this path must have even length and the last edge of the path would be from the matching. This implies if an alternating path from $u$ to a saturated vertex $s$ has even length, then all paths from $u$ to $s$ have even length, and they must all use as their last edge the only matched edge out of $s$. However, in the non-bipartite graph on the right, this is not the case. For example, we can find an alternating path of length 4 from $u$ to $z$, namely $u x y w z$, but another alternating path from $u$ to $z$ exists of length 3 where the last edge is not matched, namely $u x y z$. This is caused by the odd cycle occurring between $y, w$, and $z$. Note that the vertex from which these two paths diverge (namely $y$ ) is entered by a matched edge $(x y)$ and has two possible unmatched edges out ( $y w$ and $y z)$. This configuration is the basis behind the blossom.

## 数学代写|图论代写graph theory代考|Chinese Postman Problem

In Section 2.1.5 we introduced eulerizations and weighted-eulerizations as a way to modify a graph to ensure an eulerian circuit exists. Recall this was useful for many applications where an exhaustive route is needed, such as snowplows, mail delivery, and 3D printing. In small examples we did not need any fancy techniques as it was fairly easy to see the solution, or to use a few instances of trial and error. The weighted-eulerization problem, commonly called the Chinese Postman Problem, originates from the Chinese mathematician Guan Meigu who proposed the problem in 1960 and was solved about a decade later by Jack Edmonds and Ellis Johnson [29][42]. Their algorithm uses both Dijkstra’s Algorithm for finding a shortest path (see Section 2.3.1) and a matching in a complete graph.
Postman Algorithm
Input: Weighted graph $G=(V, E, w)$.
Steps:

1. Find the set $S$ of odd vertices in $G$.
2. Form the complete graph $K_{n}$ where $n=|S|$.
3. For each distinct pair $x, y \in S$, find the shortest path $P_{x y}$ and its total weight $w\left(P_{x y}\right)$.
4. Define the weight of the edge $x y$ in $K_{n}$ to be $\left.w(x y)=w\left(P_{x y}\right)\right)$
5. Find a perfect matching $M$ of $K_{n}$ of least total weight.
6. For each edge $e=x y \in M$, duplicate the edges of $P_{x, y}$ corresponding to the shortest path from $x$ to $y$, creating $G^{*}$.
7. Find an eulerian circuit of $G^{*}$.
Output: Optimal weighted-eulerization of $G$.

## 数学代写|图论代写GRAPH THEORY代考|CHINESE POSTMAN PROBLEM

Postman 算法

1. 找到集合小号奇数个顶点G.
2. 形成完整的图表ķn在哪里n=|小号|.
3. 对于每个不同的对X,是∈小号, 找到最短路径磷X是及其总重量在(磷X是).
4. 定义边的权重X是在ķn成为在(X是)=在(磷X是))
5. 找到一个完美的匹配米的ķn总重量最小的。
6. 对于每条边和=X是∈米, 复制边缘磷X,是对应于最短路径X到是, 创建G∗.
7. 求欧拉回路G∗.
输出：最优加权欧拉化G.

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。