## 数学代写|组合优化代写Combinatorial optimization代考|CSC205

statistics-lab™ 为您的留学生涯保驾护航 在代写组合优化Combinatorial optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合优化Combinatorial optimization代写方面经验极为丰富，各种代写组合优化Combinatorial optimization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合优化代写Combinatorial optimization代考|Rectilinear Minimum Spanning Tree

Consider two points $A=\left(x_1, y_1\right)$ and $B=\left(x_2, y_2\right)$ in the plane. The rectilinear distance of $A$ and $B$ is defined by
$$d(A, B)=\left|x_1-x_2\right|+\left|y_1-y_2\right| .$$
The rectilinear plane is the plane with the rectilinear distance, denoted by $L_1$-plane. In this section, we study the following problem.

Problem 2.2.1 (Rectilinear Minimum Spanning Tree) Given $n$ points in the rectilinear plane, compute the minimum spanning tree on those $n$ given points.
In Chap. 1, we already present Kruskal algorithm which can compute a minimum spanning tree within $O(m \log n)$ time. In this section, we will improve this result by showing that the rectilinear minimum spanning tree can be computed in $O(n \log n)$ time. To do so, we first study an interesting problem as follows.

Problem 2.2.2 (All Northeast Nearest Neighbors) Consider a set $P$ of $n$ points in the rectilinear plane. For each $A=\left(x_A, y_A\right) \in P$, another point $B=\left(x_B, y_B\right) \in P$ is said to lie in northeast (NE) area of $A$ if $x_A \leq x_B$ and $y_A \leq y_B$, but $A \neq B$. Furthermore, $B$ is the NE nearest neighbor of $A$ if $B$ has the shortest distance from A among all points lying in the $N E$ area of $A$. This problem is required to compute the NE nearest neighbor for every point in $P$. (The NE nearest neighbor of a point $A$ is “none” if no given point lies in the northeast area of $A$.)

Let us design a divide-and-conquer algorithm to solve this problem. For simplicity of description, assume all $n$ points have distinct $x$-coordinates and distinct $y$-coordinates. Now, we bisect $n$ points by a vertical line $L$. Let $P_l$ be the set of points lying on the left side of $L$ and $P_r$ the set of points lying on the right side of $L$. Suppose we already solve the all NE nearest neighbors problem on input point sets $P_l$ and $P_r$, respectively. Let us discuss how to combine solutions for two subproblems into a solution for all NE nearest neighbors on $P$.

## 数学代写|组合优化代写Combinatorial optimization代考|Fibonacci Search

Consider a sequence of $n$ distinct integers which are stored in an array $A[1 . . n]$. An element $A[i]$ is a local maximum if $A[i-1]A[i+1]$ for $1S[i+1]$ for $i=1$, and $A[i-1]<A[i]$ for $i=n$. The sequence $A[1 . . n]$ is said to be bitonic if it contains exactly one local maximum, which is actually the global maximum one. Consider the following problem.

Problem 2.3.1 (Maximum Element in Bitonic Sequence) Given a sequence $A[1 . . n]$ of $n$ distinct integers, find the maximum element.
The problem can be solved by the following lemma.
Lemma 2.3.2 Assume $1 \leq iA[j]$, then $A[1 . . j-1]$ must contain a local maximum.
Proof First, assume $A[i]A[j+1]$. In this case, if none of $A[j], A[j-1], \ldots, A[i-1]$ is a local maximum, then $A[j]<A[j-1]<\cdots<A[i]$, contradicting to $A[i]<A[j]$
Similarly, we can show the second statement.
For $n \geq 4$, we can choose $i$ and $j$ such that $1 \leq i<j \leq n, i \geq n / 3$, and $n-j+1 \geq n / 3$. With such $i$ and $j$, for each comparison, the sequence can be cut off at least one third. Therefore, the maximum element can be found within $O(\log n)$ comparisons.

Next, we consider a situation that $A[i]=f(i)$, that is, $A[i]$ has to be obtained through evaluation of a function $f(i)$. Therefore, we want to find the maximum element with the minimum number of evaluations. In this situation, $i$ and $j$ will be selected based on a rule with Fibonacci number $F_i$ defined as follows:
$$F_0=F_1=1, F_i=F_{i-2}+F_{i-1} \text { for } i \geq 2$$

# 组合优化代写

## 数学代写|组合优化代写Combinatorial optimization代考|Rectilinear Minimum Spanning Tree

$$d(A, B)=\left|x_1-x_2\right|+\left|y_1-y_2\right| .$$

## 数学代写|组合优化代写Combinatorial optimization代考|Fibonacci Search

$A[1 . n n]$ 如果它恰好包含一个局部最大值，而实际上是全局最大值，则称它是双调的。考虑以 下问题。

$$F_0=F_1=1, F_i=F_{i-2}+F_{i-1} \text { for } i \geq 2$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|组合优化代写Combinatorial optimization代考|COMP567

statistics-lab™ 为您的留学生涯保驾护航 在代写组合优化Combinatorial optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合优化Combinatorial optimization代写方面经验极为丰富，各种代写组合优化Combinatorial optimization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合优化代写Combinatorial optimization代考|Data Structure

A data structure is a data storage format which is organized and managed to have efficient access and modification. Each data structure has several standard operations. They are building bricks to construct algorithms. The data structure plays an important role in improving efficiency of algorithms. For example, we may introduce a data structure “Disjoint Sets” to improve Kruskal algorithm.

Consider a collection of disjoint sets. For each set $S$, let First $(S)$ denote the first node in set $S$. For each element $x$ in set $S$, denote First $(x)=\operatorname{First}(S)$. Define three operations as follows:
Make-Set $(x)$ creates a new set containing only $x$.
Union $(x, y)$ unions sets $S_x$ and $S_y$ containing $x$ and $y$, respectively, into $S_x \cup S_y$,
Moreover, set
$\operatorname{First}\left(S_x \cup S_y\right)= \begin{cases}\operatorname{First}\left(S_x\right) & \text { if }\left|S_x\right| \geq\left|S_y\right|, \ \operatorname{First}\left(S_y\right) & \text { otherwise. }\end{cases}$
Find-Set $(x)$ returns First $\left(S_x\right)$ where $S_x$ is the set containing element $x$.
With this data structure, Kruskal algorithm can be modified as follows.

## 数学代写|组合优化代写Combinatorial optimization代考|Algorithms with Self-Reducibility

There exist a large number of algorithms in which the problem is reduced to several subproblems, each of which is the same problem on a smaller-size input. Such a problem is said to have the self-reducibility, and the algorithm is said to be with self-reducibility.

For example, consider sorting problem again. Suppose input contains $n$ numbers. We may divide these $n$ numbers into two subproblems. One subproblem is the sorting problem on $\lfloor n / 2\rfloor$ numbers, and the other subproblem is the sorting problem on $\lceil n / 2\rceil$ numbers. After completely sorting each subproblem, combine two sorted sequences into one. This idea will result in a sorting algorithm, called the merge sort. The pseudocode of this algorithm is shown in Algorithm 1.

The main body calls a procedure. This procedure contains two self-calls, which means that the merge sort is a recursive algorithm, that is, the divide will continue until each subproblem has input of single number. Then this procedure employs another procedure (Merge) to combine solutions of subproblems with smaller inputs into subproblems with larger inputs. This computation process on input ${5,2,7,4,6,8,1,3}$ is shown in Fig. 2.1.

Note that the running time of procedure Merge at each level is $O(n)$. Let $t(n)$ be the running time of merge sort on input of size $n$. By the recursive structure, we can obtain that $t(1)=0$ and
$$t(n)=t(\lfloor n / 2\rfloor)+t(\lceil n / 2\rceil)+O(n) .$$
Suppose
$$t(n) \leq 2 \cdot t(\lceil n / 2\rceil)+c \cdot n$$
for some positive constant $c$.

# 组合优化代写

## 数学代写|组合优化代写Combinatorial optimization代考|Data Structure

Make-Set $(x)$ 创建一个仅包含的新集合 $x$.

$\operatorname{First}\left(S_x \cup S_y\right)=\left{\operatorname{First}\left(S_x\right) \quad\right.$ if $\left|S_x\right| \geq\left|S_y\right|, \operatorname{First}\left(S_y\right)$ otherwise. 查找集 $(x)$ 首先返回 $\left(S_x\right)$ 在哪里 $S_x$ 是包含元素的集合 $x$.

## 数学代写|组合优化代写Combinatorial optimization代考|Algorithms with Self-Reducibility

$$t(n)=t(\lfloor n / 2\rfloor)+t(\lceil n / 2\rceil)+O(n) .$$

$$t(n) \leq 2 \cdot t(\lceil n / 2\rceil)+c \cdot n$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|组合优化代写Combinatorial optimization代考|MTH5107

statistics-lab™ 为您的留学生涯保驾护航 在代写组合优化Combinatorial optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合优化Combinatorial optimization代写方面经验极为丰富，各种代写组合优化Combinatorial optimization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合优化代写Combinatorial optimization代考|Optimal and Approximation Solutions

Let us show an optimality condition for the minimum spanning tree.
Theorem 1.2.1 (Path Optimality) A spanning tree $T^*$ is a minimum spanning tree if and only if it satisfies the following condition:

Path Optimality Condition For every edge $(u, v)$ not in $T^$, there exists a path $p$ in $T^$, connecting $u$ and $v$, and moreover, $c(u, v) \geq c(x, y)$ for every edge $(x, y)$ in path $p$.

Proof Suppose, for contradiction, that $c(u, v)<c(x, y)$ for some edge $(x, y)$ in the path $p$. Then $T^{\prime}=\left(T^* \backslash(x, y)\right) \cup(u, v)$ is a spanning tree with cost less than $c\left(T^\right)$, contradicting the minimality of $T^$.

Conversely, suppose that $T^$ satisfies the path optimality condition. Let $T^{\prime}$ be a minimum spanning tree such that among all minimum spanning tree, $T^{\prime}$ is the one with the most edges in common with $T^$. Suppose, for contradiction, that $T^{\prime} \neq T^$. We claim that there exists an edge $(u, v) \in T^$ such that the path in $T^{\prime}$ between $u$ and $v$ contains an edge $(x, y)$ with length $c(x, y) \geq c(u, v)$. If this claim is true, then $\left(T^{\prime} \backslash(x, y)\right) \cup(u, v)$ is still a minimum spanning tree, contradicting the definition of $T^{\prime}$.

Now, we show the claim by contradiction. Suppose the claim is not true. Consider an edge $\left(u_1, v_1\right) \in T^* \backslash T^{\prime}$. the path in $T^{\prime}$ connecting $u_1$ and $v_1$ must contain an edge $\left(x_1, y_1\right)$ not in $T^$. Since the claim is not true, we have $c\left(u_1, v_1\right)$ connecting $x_1$ and $y_1$, which must contain an edge $\left(u_2, v_2\right) \notin$ $T^{\prime}$. Since $T^$ satisfies the path optimality condition, we have $c\left(x_1, y_1\right) \leq c\left(u_2, v_2\right)$. Hence, $c\left(u_1, v_1\right)$ such that $c\left(u_1, v_2\right)<c\left(u_2, v_2\right)<c\left(u_3, v_3\right)<\cdots$, contradicting the finiteness of $T^*$.

## 数学代写|组合优化代写Combinatorial optimization代考|Running Time

The most important measure of quality for algorithms is the running time. However, for the same algorithm, it may take different times when we run it in different computers. To give a uniform standard, we have to get an agreement that runs algorithms in a theoretical computer model. This model is the multi-tape Turing machine which has been accepted by a very large population. Based on the Turing machine, the theory of computational complexity has been built up. We will touch this part of theory in Chap. 8.

But, we will use RAM model to evaluate the running time for algorithms throughout this book except Chap. 8. In RAM model, assume that each line of pseudocode requires a constant time. For example, the running time of insertion sort is calculated in Fig. 1.6.

Actually, RAM model and Turing machine model are closely related. The running time estimated based on these two models is considered to be close enough. However, they are sometimes different in estimation of running time. For example, the following is a piece of pseudocode.
for $i=1$ to $n$
do assign First $(i) \leftarrow i$
end-for
According to RAM model, the running time of this piece is $O(n)$. However, based on the Turing machine, the running time of this piece is $O(n \log n)$ because the assigned value has to be represented by a string with $O(\log n)$ symbols.

Theoretically, a constant factor is often ignored. For example, we usually say that the running time of insertion sort is $O\left(n^2\right)$ instead of giving the specific quadratic function with respect to $n$. Here $f(n)=O(g(n))$ means that there exist constants $c>0$ and $n_0>0$ such that
$$f(n) \leq c \cdot g(n) \text { for } n \geq n_0$$

# 组合优化代写

## 数学代写|组合优化代写Combinatorial optimization代考|Fibonacci Search

$$F_0=F_1=1, F_i=F_{i-2}+F_{i-1} \text { for } i \geq 2$$

## 数学代写|组合优化代写Combinatorial optimization代考|Dynamic Programming

$$F_0=0, F_1=1 \text {, and } F_i=F_{i-1}+F_{i-2} .$$

$$\operatorname{cost}(T)=\sum_{i=1}^n a_i D_i .$$

$\operatorname{sum}(i, j)=a_i+a_{i+1}+\cdots+a_j$. 然后
$$\operatorname{cost}(T(i, j))=\min _{i \leq k<j} \cos t(T(i, k))+\cos t(T(k+1, j))+\operatorname{sum}(i, j)$$

$$\operatorname{sum}(i, j)=\left{a_i \quad \text { if } i=j a_i+\operatorname{sum}(i+1, j) \quad \text { if } i<j .\right.$$

running time $=($ 子问题的数量 $) \times($ 递归的计算时间 $)$ 。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|组合优化代写Combinatorial optimization代考|ORIE6334

statistics-lab™ 为您的留学生涯保驾护航 在代写组合优化Combinatorial optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合优化Combinatorial optimization代写方面经验极为丰富，各种代写组合优化Combinatorial optimization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合优化代写Combinatorial optimization代考|Fibonacci Search

Consider a sequence of $n$ distinct integers which are stored in an array $A[1 . . n]$. An element $A[i]$ is a local maximum if $A[i-1]A[i+1]$ for $1S[i+1]$ for $i=1$, and $A[i-1]<A[i]$ for $i=n$. The sequence $A[1 . . n]$ is said to be bitonic if it contains exactly one local maximum, which is actually the global maximum one. Consider the following problem.

Problem 2.3.1 (Maximum Element in Bitonic Sequence) Given a sequence $A[1 . . n]$ of $n$ distinct integers, find the maximum element.
The problem can be solved by the following lemma.
Lemma 2.3.2 Assume $1 \leq iA[j]$, then $A[1 . . j-1]$ must contain a local maximum.
Proof First, assume $A[i]A[j+1]$. In this case, if none of $A[j], A[j-1], \ldots, A[i-1]$ is a local maximum, then $A[j]<A[j-1]<\cdots<A[i]$, contradicting to $A[i]<A[j]$.
Similarly, we can show the second statement.
For $n \geq 4$, we can choose $i$ and $j$ such that $1 \leq i<j \leq n, i \geq n / 3$, and $n-j+1 \geq n / 3$. With such $i$ and $j$, for each comparison, the sequence can be cut off at least one third. Therefore, the maximum element can be found within $O(\log n$ ) comparisons.

Next, we consider a situation that $A[i]=f(i)$, that is, $A[i]$ has to be obtained through evaluation of a function $f(i)$. Therefore, we want to find the maximum element with the minimum number of evaluations. In this situation, $i$ and $j$ will be selected based on a rule with Fibonacci number $F_i$ defined as follows:
$$F_0=F_1=1, F_i=F_{i-2}+F_{i-1} \text { for } i \geq 2$$

## 数学代写|组合优化代写Combinatorial optimization代考|Dynamic Programming

Let us first study several examples and start from a simpler one.
Example 3.1.1 (Fibonacci Number) Fibonacci number $F_i$ for $i=0,1, \ldots$ is defined by
$$F_0=0, F_1=1 \text {, and } F_i=F_{i-1}+F_{i-2} .$$
The computational process can be considered as a dynamic programming with self-reducibility structure as shown in Fig. 3.1.

Example 3.1.2 (Labeled Tree) Let $a_1, a_2, \ldots, a_n$ be a sequence of $n$ positive integers. A labeled tree for this sequence is a binary tree $T$ of $n$ leaves named $v_1, v_2, \ldots, v_n$ from left to right. We label $v_i$ by $a_i$ for all $i, 1 \leq i \leq n$. Let $D_i$

be the length of the path from $v_i$ to the root of $T$. The $\operatorname{cost}$ of $T$ is defined by
$$\operatorname{cost}(T)=\sum_{i=1}^n a_i D_i .$$
The problem is to construct a labeled tree $T$ to minimize the cost $\operatorname{cost}(T)$ for a given sequence of positive integers $a_1, a_2, \ldots, a_n$.

Let $T(i, j)$ be the optimal labeled tree for subsequence $\left{a_i, a_{i+1}, \ldots, a_j\right}$ and $\operatorname{sum}(i, j)=a_i+a_{i+1}+\cdots+a_j$. Then
$$\operatorname{cost}(T(i, j))=\min _{i \leq k<j}{\cos t(T(i, k))+\cos t(T(k+1, j))}+\operatorname{sum}(i, j)$$
where
$$\operatorname{sum}(i, j)= \begin{cases}a_i & \text { if } i=j \ a_i+\operatorname{sum}(i+1, j) & \text { if } i<j .\end{cases}$$
As shown in Fig. 3.2, there are $1+2+\cdots+n=\frac{n(n+1)}{2}$ subproblems $T(i, j)$ in the table. From recursive formula, it can be seen that solution of each subproblem $T(i, j)$ can be computed in $O(n)$ time. Therefore, this dynamic programming runs totally in $O\left(n^3\right)$ time.

Actually, the running time of a dynamic programming is often estimated by the following formula:
running time $=$ (number of subproblems $) \times($ computing time of recursion).

# 组合优化代写

## 数学代写|组合优化代写Combinatorial optimization代考|Fibonacci Search

$$F_0=F_1=1, F_i=F_{i-2}+F_{i-1} \text { for } i \geq 2$$

## 数学代写|组合优化代写Combinatorial optimization代考|Dynamic Programming

$$F_0=0, F_1=1 \text {, and } F_i=F_{i-1}+F_{i-2} .$$

$$\operatorname{cost}(T)=\sum_{i=1}^n a_i D_i .$$

$\operatorname{sum}(i, j)=a_i+a_{i+1}+\cdots+a_j$. 然后
$$\operatorname{cost}(T(i, j))=\min _{i \leq k<j} \cos t(T(i, k))+\cos t(T(k+1, j))+\operatorname{sum}(i, j)$$

$$\operatorname{sum}(i, j)=\left{a_i \quad \text { if } i=j a_i+\operatorname{sum}(i+1, j) \quad \text { if } i<j .\right.$$

running time $=($ 子问题的数量 $) \times($ 递归的计算时间 $)$ 。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|组合优化代写Combinatorial optimization代考|CSC205

statistics-lab™ 为您的留学生涯保驾护航 在代写组合优化Combinatorial optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合优化Combinatorial optimization代写方面经验极为丰富，各种代写组合优化Combinatorial optimization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合优化代写Combinatorial optimization代考|Algorithms with Self-Reducibility

There exist a large number of algorithms in which the problem is reduced to several subproblems, each of which is the same problem on a smaller-size input. Such a problem is said to have the self-reducibility, and the algorithm is said to be with self-reducibility.

For example, consider sorting problem again. Suppose input contains $n$ numbers. We may divide these $n$ numbers into two subproblems. One subproblem is the sorting problem on $\lfloor n / 2\rfloor$ numbers, and the other subproblem is the sorting problem on $\lceil n / 2\rceil$ numbers. After completely sorting each subproblem, combine two sorted sequences into one. This idea will result in a sorting algorithm, called the merge sort. The pseudocode of this algorithm is shown in Algorithm 1.

The main body calls a procedure. This procedure contains two self-calls, which means that the merge sort is a recursive algorithm, that is, the divide will continue until each subproblem has input of single number. Then this procedure employs another procedure (Merge) to combine solutions of subproblems with smaller

$$T(n)=2 \cdot T(\lceil n / 2\rceil)+c \cdot n$$
By induction, we can prove that
$$t(n) \leq T(n) \text { for all } n \geq 1$$
For base step, $t(1)=0=T(1)$. For induction step,
\begin{aligned} t(n) & \leq 2 \cdot t(\lceil n / 2\rceil)+c \cdot n \ & \leq 2 \cdot T(\lceil n / 2\rceil)+c \cdot n \quad \text { (by induction hypothesis) } \ & =T(n) . \end{aligned}
Now, let us discuss how to solve recursive equation about $T(n)$. Usually, we use two stages. In the first stage, we consider special numbers $n=2^k$ and employ the recursive tree to find $T\left(2^k\right)$ (Fig. $\left.2.2\right)$, that is,
\begin{aligned} T\left(2^k\right) & =2 \cdot T\left(2^{k-1}\right)+c \cdot 2^k \ & =2 \cdot\left(2 \cdot T\left(2^{k-2}\right)+c \cdot 2^{k-1}\right)+c \cdot 2^k \ & =\ldots \ & =2^k T(1)+k c \cdot 2^k \end{aligned}

## 数学代写|组合优化代写Combinatorial optimization代考|Rectilinear Minimum Spanning Tree

Consider two points $A=\left(x_1, y_1\right)$ and $B=\left(x_2, y_2\right)$ in the plane. The rectilinear distance of $A$ and $B$ is defined by
$$d(A, B)=\left|x_1-x_2\right|+\left|y_1-y_2\right| .$$
The rectilinear plane is the plane with the rectilinear distance, denoted by $L_1$-plane. In this section, we study the following problem.

Problem 2.2.1 (Rectilinear Minimum Spanning Tree) Given $n$ points in the rectilinear plane, compute the minimum spanning tree on those $n$ given points.
In Chap. 1, we already present Kruskal algorithm which can compute a minimum spanning tree within $O(m \log n)$ time. In this section, we will improve this result by showing that the rectilinear minimum spanning tree can be computed in $O(n \log n)$ time. To do so, we first study an interesting problem as follows.

Problem 2.2.2 (All Northeast Nearest Neighbors) Consider a set $P$ of $n$ points in the rectilinear plane. For each $A=\left(x_A, y_A\right) \in P$, another point $B=\left(x_B, y_B\right) \in P$ is said to lie in northeast (NE) area of $A$ if $x_A \leq x_B$ and $y_A \leq y_B$, but $A \neq B$. Furthermore, $B$ is the NE nearest neighbor of $A$ if $B$ has the shortest distance from A among all points lying in the NE area of A. This problem is required to compute the NE nearest neighbor for every point in $P$. (The NE nearest neighbor of a point A is “none” if no given point lies in the northeast area of A.)

Let us design a divide-and-conquer algorithm to solve this problem. For simplicity of description, assume all $n$ points have distinct $x$-coordinates and distinct $y$-coordinates. Now, we bisect $n$ points by a vertical line $L$. Let $P_l$ be the set of points lying on the left side of $L$ and $P_r$ the set of points lying on the right side of $L$. Suppose we already solve the all NE nearest neighbors problem on input point sets $P_l$ and $P_r$, respectively. Let us discuss how to combine solutions for two subproblems into a solution for all NE nearest neighbors on $P$.

For point $A$ in $P_r$, the NE nearest neighbor in $P_r$ is also the NE nearest neighbor in $P$. However, for point $A$ in $P_l$, the NE nearest neighbor in $P_l$ may not be the NE nearest neighbor in $P$. Actually, let $B_1$ denote the NE nearest neighbor of $A$ in $P_l$ and $B_r$ the NE nearest neighbor of $A$ for $B_2$ in $P_r$. Then, if $d\left(A, B_1\right) \leq d\left(A, B_2\right)$, then the NE nearest neighbor of $A$ in $P$ is $B_1$; otherwise, it is $B_2$. Therefore, to complete the combination task, it is sufficient to compute the NE nearest neighbors in $P_r$ for all points in $P_l$. We will show that this computation takes $O(n)$ time. To do so, let us first show a lemma.

# 组合优化代写

## 数学代写|组合优化代写Combinatorial optimization代考|Algorithms with Self-Reducibility

$$T(n)=2 \cdot T(\lceil n / 2\rceil)+c \cdot n$$

$$t(n) \leq T(n) \text { for all } n \geq 1$$

$$t(n) \leq 2 \cdot t(\lceil n / 2\rceil)+c \cdot n \quad \leq 2 \cdot T(\lceil n / 2\rceil)+c \cdot n \quad \text { (by induction hypothesis) }=T(n) .$$

$$T\left(2^k\right)=2 \cdot T\left(2^{k-1}\right)+c \cdot 2^k \quad=2 \cdot\left(2 \cdot T\left(2^{k-2}\right)+c \cdot 2^{k-1}\right)+c \cdot 2^k=\ldots \quad=2^k T(1)$$

## 数学代写|组合优化代写Combinatorial optimization代考|Rectilinear Minimum Spanning Tree

$$d(A, B)=\left|x_1-x_2\right|+\left|y_1-y_2\right|$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|组合优化代写Combinatorial optimization代考|APM6664

statistics-lab™ 为您的留学生涯保驾护航 在代写组合优化Combinatorial optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合优化Combinatorial optimization代写方面经验极为丰富，各种代写组合优化Combinatorial optimization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合优化代写Combinatorial optimization代考|What Is Combinatorial Optimization

The aim of combinatorial optimization is to find an optimal object from a finite set of objects. Those candidate objects are called feasible solutions, while the optimal one is called an optimal solution. For example, consider the following problem.
Problem 1.1.1 (Minimum Spanning Tree) Given a connected graph $G=(V, E)$ with nonnegative edge weight $c: E \rightarrow R_{+}$, find a spanning tree with minimum total weight, where “spanning” means that all nodes are included in the tree and hence a spanning tree interconnects all nodes in V. An example is as shown in Fig. 1.1.
Clearly, the set of all spanning trees is finite, and the aim of this problem is to find one with minimum total weight from this set. Each spanning tree is a feasible solution, and the optimal solution is the spanning tree with minimum total weight, which is also called the minimum spanning tree. Therefore, this is a combinatorial optimization problem.

A combinatorial optimization problem may have more than one optimal solution. For example, in Fig. 1.1, there are two spanning trees with minimum total length. (The second one can be obtained by using edge $(e, f)$ to replace edge $(d, f)$.) Therefore, by the optimal solution as mentioned in the above, it means a general member in the class of optimal solutions.

The combinatorial optimization is a proper subfield of discrete optimization. In fact, there exist some problems in discrete optimization, which do not belong to combinatorial optimization. For example, consider the integer programming. It always belongs to discrete optimization. However, when feasible domain is infinite, it does not belong to combinatorial optimization. But such a difference is not recognized very well in the literature. Actually, if a paper on lattice-point optimization is submitted to Journal of Combinatorial Optimization, then usually, it will not be rejected due to out of scope.

In view of methodologies, combinatorial optimization and discrete optimization have very close relationship. For example, to prove NP-hardness of integer programming, we need to cut its infinitely large feasible domain into a finite subset containing optimal solution (see Chap. 8 for details), i.e., transform it into a combinatorial optimization problem.
Geometric optimization is another example. Consider the following problem.

## 数学代写|组合优化代写Combinatorial optimization代考|Optimal and Approximation Solutions

Let us show an optimality condition for the minimum spanning tree.
Theorem 1.2.1 (Path Optimality) A spanning tree $T^*$ is a minimum spanning tree if and only if it satisfies the following condition:

Path Optimality Condition For every edge $(u, v)$ not in $T^$, there exists a path $p$ in $T^$, connecting $u$ and $v$, and moreover, $c(u, v) \geq c(x, y)$ for every edge $(x, y)$ in path $p$.

Proof Suppose, for contradiction, that $c(u, v)<c(x, y)$ for some edge $(x, y)$ in the path $p$. Then $T^{\prime}=\left(T^* \backslash(x, y)\right) \cup(u, v)$ is a spanning tree with cost less than $c\left(T^\right)$, contradicting the minimality of $T^$.

Conversely, suppose that $T^$ satisfies the path optimality condition. Let $T^{\prime}$ be a minimum spanning tree such that among all minimum spanning tree, $T^{\prime}$ is the one with the most edges in common with $T^$. Suppose, for contradiction, that $T^{\prime} \neq T^$. We claim that there exists an edge $(u, v) \in T^$ such that the path in $T^{\prime}$ between $u$ and $v$ contains an edge $(x, y)$ with length $c(x, y) \geq c(u, v)$. If this claim is true, then $\left(T^{\prime} \backslash(x, y)\right) \cup(u, v)$ is still a minimum spanning tree, contradicting the definition of $T^{\prime}$.

Now, we show the claim by contradiction. Suppose the claim is not true. Consider an edge $\left(u_1, v_1\right) \in T^* \backslash T^{\prime}$. the path in $T^{\prime}$ connecting $u_1$ and $v_1$ must contain an edge $\left(x_1, y_1\right)$ not in $T^$. Since the claim is not true, we have $c\left(u_1, v_1\right)$ connecting $x_1$ and $y_1$, which must contain an edge $\left(u_2, v_2\right) \notin$ $T^{\prime}$. Since $T^$ satisfies the path optimality condition, we have $c\left(x_1, y_1\right) \leq c\left(u_2, v_2\right)$. Hence, $c\left(u_1, v_1\right)$ such that $c\left(u_1, v_2\right)<c\left(u_2, v_2\right)<c\left(u_3, v_3\right)<\cdots$, contradicting the finiteness of $T^*$.

From this algorithm, we see that it is not hard to find the optimal solution for the minimum spanning tree problem. If every combinatorial optimization problem likes the minimum spanning tree, then we would be very happy to find optimal solution for it. Unfortunately, there exist a large number of problems that it is unlikely to be able to compute their optimal solution efficiently. For example, consider the following problem.

# 组合优化代写

1.1 所示。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|组合优化代写Combinatorial optimization代考|CSC205

statistics-lab™ 为您的留学生涯保驾护航 在代写组合优化Combinatorial optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合优化Combinatorial optimization代写方面经验极为丰富，各种代写组合优化Combinatorial optimization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合优化代写Combinatorial optimization代考|Polyhedra and Linear Programming

Definition 1. Let $x_1, x_2, \ldots, x_m$ be points in $\mathbb{R}^n$. Let $x=\sum_{i=1}^m \lambda_i x_i$, where each $\lambda_i \in \mathbb{R}, 1 \leq i \leq m$ is a scalar. Then, $x$ is said to be a(n)

1. Linear combination (of $x_i, 1 \leq i \leq m$ ) for arbitrary scalars $\lambda_i$.
2. Affine combination if $\sum_i \lambda_i=1$.
3. Conical combination if $\lambda_i \geq 0$.
4. Convex combination if $\sum \lambda_i=1$ and $\lambda_i \geq 0$ (affine and also canonical).
In the following definitions and propositions, unless otherwise stated, it will be assumed that $x_1, x_2, \ldots, x_m$ are points in $\mathbb{R}^n$ and $\lambda_1, \lambda_2, \ldots, \lambda_m$ are scalars in $\mathbb{R}$.
Definition 2. $x_1, x_2, \ldots, x_m$ are said to be linearly independent if $\sum_{i=1}^m \lambda_i x_i=0 \Rightarrow \forall i \in[m] \lambda_i=0$.
Definition 3. $x_1, x_2, \ldots, x_m$ are said to be affinely independent if the vectors $\left(x_i-x_1\right), i=2, \ldots, m$ are linearly independent, or equivalently if $\sum_{i=1}^m \lambda_i x_i=0$ and $\sum_{i=1}^m \lambda_i=0 \Rightarrow \forall i \in[m] \lambda_i=0$.
The following proposition is easy to check and the proof is left as an exercise to the reader.
Proposition 4. $x_1, x_2, \ldots, x_m$ are affinely independent if and only if the vectors $\left(\begin{array}{c}x_i \ 1\end{array}\right), i=1,2, \ldots, m$, are linearly independent in $\mathbb{R}^{m+1}$.

A set $X \subseteq \mathbb{R}^n$ is said to be a(n) subspace [affine set, cone set, convex set] if it is closed under linear [affine, conical, convex] combinations. Note that an affine set is a translation of a subspace. Given $X \subseteq$ $\mathbb{R}^n$, we let $\operatorname{Span}(X), \operatorname{Af}(X)$, Cone $(X)$, and Convex $(X)$ denote the closures of $X$ under linear, affine, conical, and convex combinations, respectively. To get an intuitive feel of the above definitions, see Figure 1.

## 数学代写|组合优化代写Combinatorial optimization代考|Fourier-Motzkin Elimination

Let $P={x \mid A x \leq b} \subseteq \mathbb{R}^n$ be a polyhedron. For $k$ in $[n]$,
we let $P^k=\left{\left(x_1, \ldots, x_{k-1}, x_{k+1}, \ldots, x_n\right) \mid\left(x_1, x_2, \ldots, x_n\right) \in P\right}$ be the projection of $P$ along the $x_k$-axis.
Theorem 13. $P^k$ is a polyhedron.
Proof. We derive a set of inequalities that describe $P^k$. We do this by considering the inequalities in $A x \leq b$ and eliminating the variables $x_k$ as follows. Partition the inequalities in $A x \leq b$ into three sets:
$$S^{+}=\left{i \in[m] \mid a_{i k}>0\right}, S^{-}=\left{i \in[m] \mid a_{i k}<0\right}, \text { and } S^0=\left{i \in[m] \mid a_{i k}=0\right} .$$
Define a new set of inequalities consisting of $S_0$ and one new inequality for each pair $(i, \ell)$ in $S^{+} \times S^{-}$:
$$a_{i k}\left(\sum_{j=1}^n a_{\ell j} x_j\right)-a_{\ell k}\left(\sum_{j=1}^n a_{i j} x_j\right) \leq a_{i k} b_{\ell}-a_{\ell k} b_i .$$
Note that the combined inequality does not have $x_k$. We now have a total of $\left|S^0\right|+\left|S^{+}\right|\left|S^{-}\right|$new inequalities. Let $P^{\prime}=\left{x^{\prime} \in \mathbb{R}^{n-1} \mid A^{\prime} x^{\prime} \leq b^{\prime}\right}$ where $A^{\prime} x^{\prime} \leq b^{\prime}$ is the new system of inequalities in variables $x_1, x_2, \ldots, x_{k-1}, x_{k+1}, \ldots, x_n$. We prove the theorem by showing that $P^k=P^{\prime}$.

We first show the easier direction: $P^k \subseteq P^{\prime}$. Consider any point $z \in P^k$. By definition of $P^k$, there exists $x \in P$ such that $A x \leq b$ and $x$ ‘s projection along $x_k$-axis is $z$. It is easy to see that $z$ satisfies the new system since the new one was obtained in a way oblivious to $x_k$, the real value of $x$ ‘s $k_{\text {th }}$ coordinate.
We now show that $P^{\prime} \subseteq P^k$. Without loss of generality, assume $k=1$. Consider any $x^{\prime}=\left(x_2, x_3, \ldots, x_n\right) \in$ $P^{\prime}$. We want to show that there exists $x_1 \in \mathbb{R}$ such that $A x \leq b$, where $x=\left(x_1, x_2, \ldots, x_n\right)$. For simple notation, define $C_i=b_i-\sum_{j=2}^n a_{i j} x_j$ for $i \in[m]$. Note that $A x \leq b$ can be rewritten as
$$a_{i 1} x_1 \leq C_i, \forall i \in[m] .$$
Observe that $x$ satisfies all inequalities consisting of $S^0$, since the new system as well includes those constraints. Thus we can refine our goal to show
$$\begin{array}{ll} \exists x_1 \text { s.t. } \quad & a_{i 1} x_1 \leq C_i, \forall i \in S^{+} \cup S^{-} . \ \Leftrightarrow \quad \max {\ell \in S^{-}} \frac{C{\ell}}{a_{\ell 1}} \leq x_1 \leq \min {i \in S^{+}} \frac{C_i}{a{i 1}} . \end{array}$$

# 组合优化代写

## 数学代写|组合优化代写Combinatorial optimization代考|Polyhedra and Linear Programming

1. 线性组合 (的 $x_i, 1 \leq i \leq m$ ) 对于任意标量 $\lambda_i$.
2. 仿射组合如果 $\sum_i \lambda_i=1$.
3. 锥形组合如果 $\lambda_i \geq 0$.
4. 凸组合如果 $\sum \lambda_i=1$ 和 $\lambda_i \geq 0$ (仿射和规范)。
在以下定义和命题中，除非另有说明，否则将假定 $x_1, x_2, \ldots, x_m$ 是点在 $\mathbb{R}^n$ 和 $\lambda_1, \lambda_2, \ldots, \lambda_m$ 是 标量在 $\mathbb{R}$.
定义 $2 。 x_1, x_2, \ldots, x_m$ 被称为线性独立的如果 $\sum_{i=1}^m \lambda_i x_i=0 \Rightarrow \forall i \in[m] \lambda_i=0$.
定义 3。 $x_1, x_2, \ldots, x_m$ 如果向量是仿射独立的 $\left(x_i-x_1\right), i=2, \ldots, m$ 是线性独立的，或者等 效地如果 $\sum_{i=1}^m \lambda_i x_i=0$ 和 $\sum_{i=1}^m \lambda_i=0 \Rightarrow \forall i \in[m] \lambda_i=0$.
下面的命题很容易检验，证明留给读者作为练习。
命题 4 。 $x_1, x_2, \ldots, x_m$ 仿射独立当且仅当向量 $\left(x_i 1\right), i=1,2, \ldots, m$, 是线性独立的 $\mathbb{R}^{m+1}$.
一套 $X \subseteq \mathbb{R}^n$ 如果它在线性 [仿射、圆雉、凸] 组合下闭合，则称其为 $a(n)$ 子空间 [仿射集、锥集、凸 集]。请注意，仿射集是子空间的平移。鉴于 $X \subseteq \mathbb{R}^n$ ，我们让Span $(X), \operatorname{Af}(X)$ ，圆锥体 $(X)$ ，和凸 $(X)$ 表示闭包 $X$ 分别在线性、仿射、圆锥和凸组合下。要直观地感受上述定义，请参见图 1。

## 数学代写|组合优化代写Combinatorial optimization代考|Fourier-Motzkin Elimination

$$a_{i k}\left(\sum_{j=1}^n a_{\ell j} x_j\right)-a_{\ell k}\left(\sum_{j=1}^n a_{i j} x_j\right) \leq a_{i k} b_{\ell}-a_{\ell k} b_i .$$

$C_i=b_i-\sum_{j=2}^n a_{i j} x_j$ 为了 $i \in[m]$. 注意 $A x \leq b$ 可以改写为
$$a_{i 1} x_1 \leq C_i, \forall i \in[m] .$$

$$\exists x_1 \text { s.t. } \quad a_{i 1} x_1 \leq C_i, \forall i \in S^{+} \cup S^{-} . \Leftrightarrow \quad \max \ell \in S^{-} \frac{C \ell}{a_{\ell 1}} \leq x_1 \leq \min i \in S^{+} \frac{C_i}{a i 1} \text {. }$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|组合优化代写Combinatorial optimization代考|APM6664

statistics-lab™ 为您的留学生涯保驾护航 在代写组合优化Combinatorial optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合优化Combinatorial optimization代写方面经验极为丰富，各种代写组合优化Combinatorial optimization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合优化代写Combinatorial optimization代考|Bipartite Matchings

Many of you have seen bipartite matchings reduced to flow. We can also treat them independently. Let $G=(X \cup Y, E)$ be a bipartite graph with bipartition given by $X, Y$. Recall that $M \subseteq E$ is a matching in a graph if no vertex in $G$ is incident to more than one edge in $M$.

A vertex cover in $G=(V, E)$ is a subset of vertices $S \subseteq V$ such that for each edge $u v \in E, u$ or $v$ is in $S$. In other words, $S$ covers all edges. We use $\nu(G)$ to indicate the cardinality of a maximum matching in $G$ and $\tau(G)$ for the cardinality of a minimum vertex cover in $G$.
Proposition 3 For every $G, \nu(G) \leq \tau(G)$.
In bipartite graphs, one has the following theorem.
Theorem 4 (König’s Theorem) If $G$ is bipartite then $\nu(G)=\tau(G)$.
The above proves that the max matching and min vertex problems (decision versions) in bipartite graphs are both in NP $\cap$ coNP . (Note that the vertex cover problem is NP -hard in general graphs.) We therefore expect a polynomial time algorithm for $\nu(G)$ and $\tau(G)$ in bipartite graphs. As you know, one can reduce matching in bipartite graphs to maxflow and König’s theorem follows from the maxflow-mincut theorem. One can also obtain a polynomial time augmenting path algorithm for matching in bipartite graphs (implicitly a maxflow algorithm) that proves König’s theorem algorithmically.

We now look at the polyhedral aspect. We can write a simple LP as a relaxation for the maximum matching in a graph $G$.
\begin{aligned} \max & \sum_{e \in E} x(e) \ \sum_{-G \delta(*)} x(e) & \leq 1 \quad \forall u \in V \ x(e) & \geq 0 \quad \forall e \in E \end{aligned}
For bipartite graphs the above LP and its dual have integral solutions since the constraint matrix is TUM. One can easily derive König’s theorem and a polynomial time algorithm from this. One also obtains a polynomial time algorithm for weighted matching (assignment problem).

## 数学代写|组合优化代写Combinatorial optimization代考|Tutte-Berge formula

We will prove the easy direction for now, i.e.,
$$\nu(G) \leq \frac{1}{2}(|V|+|U|-o(G-U)) \quad \forall U \subseteq V$$
To see this, the number of unmatched vertices is at least $o(G-U)-|U|$, since each odd component in $G-U$ needs a vertex in $U$. Hence
$$\nu(G) \leq \frac{|V|}{2}-\frac{o(G-U)-|U|}{2} \leq \frac{1}{2}(|V|+|U|-o(G-U))$$
Corollary 5 (Tutte’s 1-factor theorem) $G$ has a perfect matching iff $\forall U \subseteq V o(G-U) \leq|U|$.
The formula shows that the decision version of matching is in NP $\cap$ coNP . Edmonds gave the first polynomial time algorithms for finding a maximum cardinality matching and also more difficult maximum weight matching problem. As a consequence of his algorithmic work, Edmonds showed the following results on matching polytopes.
Consider the following polytope:
\begin{aligned} \sum_{e \in \delta(v)} x(e) & \leq 1 \quad \forall v \in V \ \sum_{e \in E[U]} x(e) & \leq\left\lfloor\frac{|U|}{2}\right\rfloor \quad \forall U,|U| \text { odd } \ 0 & \leq x(e) \quad \forall e \in E \end{aligned}
Edmonds showed that the vertices of the above polytope are exactly the characteristic vectors of the matchings of $G$. Observe that the polytope has an exponential number of constraints. One can ask whether this description of the matching polytope useful. Clearly if one takes the convex hull of the matchings of $G$, one obtains a polytope in $\mathbb{R}^E$; one can do this for any combinatorial optimization problem and in general one gets an exponential number of inequalities.

# 组合优化代写

## 数学代写|组合优化代写Combinatorial optimization代考|Bipartite Matchings

$$\max \sum_{e \in E} x(e) \sum_{-G \delta(*)} x(e) \quad \leq 1 \quad \forall u \in V x(e) \geq 0 \quad \forall e \in E$$

## 数学代写|组合优化代写Combinatorial optimization代考|Tutte-Berge formula

$$\nu(G) \leq \frac{1}{2}(|V|+|U|-o(G-U)) \quad \forall U \subseteq V$$

$$\nu(G) \leq \frac{|V|}{2}-\frac{o(G-U)-|U|}{2} \leq \frac{1}{2}(|V|+|U|-o(G-U))$$

$$\sum_{e \in \delta(v)} x(e) \leq 1 \quad \forall v \in V \sum_{e \in E[U]} x(e) \quad \leq\left\lfloor\frac{|U|}{2}\right\rfloor \quad \forall U,|U| \text { odd } 0 \leq x(e) \quad \forall e \in E$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|组合优化代写Combinatorial optimization代考|COMP567

statistics-lab™ 为您的留学生涯保驾护航 在代写组合优化Combinatorial optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合优化Combinatorial optimization代写方面经验极为丰富，各种代写组合优化Combinatorial optimization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合优化代写Combinatorial optimization代考|Introduction and Motivation

Roughly speaking, an optimization problem has the following outline: given an instance of the problem, find the “best” solution among all solutions to the given instance. We will be mostly interested in discrete optimization problems where the instances and the solution set for each instance is from a discrete set. This is in contrast to continuous optimization where the input instance and the solution set for an instance can come from a continuous domain. Some of these distinctions are not as clear-cut as linear programming shows.

We assume familiarity with the computational complexity classes $\mathbf{P}, \mathbf{N P}$, coNP . In this class we are mainly interested in polynomial time solvable “combinatorial” optimization problems. Combinatorial optimization problems are a subset of discrete optimization problems although there is no formal definition for them. A typical problem in combinatorial optimization has a ground set $E$ of objects and solutions correspond to some subsets of $2^E$ (the power set of $E$ ) and a typical goal is to find either a maximum or minimum weight solution for some given weights on $E$. For example, in the spanning tree problem, the ground set $E$ is the set of edges of a graph $G=(V, E)$ and the solutions are subsets of $E$ that correspond to spanning trees in $G$.

We will be interested in NP optimization problems – NPO problems for short. Formally, a problem $Q$ is a subset of $\Sigma^$, where $\Sigma$ is a finite alphabet such as binary. Each string $I$ in $Q$ is an instance of $Q$. For a string $x$ we use $|x|$ to denote its length. We say that $Q$ is an NPO problem if the following hold: (i) for each $x \in \Sigma^$ there is a polynomial time algorithm that can check if $x \in Q$, i.e., if $x$ is a valid instance of $Q$
(ii) for each instance $I$ there is a set $\operatorname{sol}(I) \subset \Sigma^$ such that (a) $\forall s \in \operatorname{sol}(I),|s|=\operatorname{poly}(|I|)$ (b) there exists a poly-time algorithm that on input $I$, s correctly outputs whether $s \in s o l(I)$ or not (iii) there is a function val : $\Sigma^ \times \Sigma^* \rightarrow \mathbb{Z}$ s.t. $\operatorname{val}(I, s)$ assigns an integer to each instance $I$ and $s \in \operatorname{sol}(I)$ and moreover val can be computed by a poly-time algorithm.

Given an NPO problem, we say it is a minimization problem if the goal is to compute, given $I \in Q$, $\arg \min {s \in \operatorname{sol}(I)} \operatorname{val}(I, s)$. It is a maximization problem if the goal is to compute arg $\max {s \in \operatorname{sol}(I)} v a l(I, s)$. A natural decision problem associated with an NPO problem (say, maximization) is: given $I$ and integer $k$, is there $s \in \operatorname{sol}(I)$ s.t. $\operatorname{val}(s, I) \geq k$.

Many problems we encounter are NPO problems. Some of them can be solved in polynomial time. It is widely believed and conjectured that $\mathbf{P} \neq \mathbf{N P}$, which would mean that there are NPO problems (in particular, those whose decision versions are NP -complete) that do not have polynomial time algorithms. Assuming $\mathbf{P} \neq \mathbf{N P}$, some important and useful questions are: What problems are in $\mathbf{P}$ ? What characterizes problems in $\mathbf{P}$ ?

## 数学代写|组合优化代写Combinatorial optimization代考|Network Flow

Given directed graph $D=(V, A)$, two distinct nodes $s, t \in V$, and arc capacities $c: A \rightarrow \mathbb{R}^{+}$, a flow is a function $f: A \rightarrow \mathbb{R}^{+}$s.t.
(i) $\sum_{a \in \delta^{-}(v)} f(a)=\sum_{a \in \delta^{+}(v)} f(a)$ for all $v \in V-{s, t}$
(ii) $0 \leq f(a) \leq c(a)$ for all $a \in A$
The value of $f$ is
$$\operatorname{val}(f)=\sum_{a \in \delta^{+}(s)} f(a)-\sum_{a \in \delta^{-}(s)} f(a)=\sum_{a \in \delta^{-}(t)} f(a)-\sum_{a \in \delta^{+}(t)} f(a)$$
The optimization problem is to maximize the flow from $s$ to $t$.
An $s$ – $t$ cut is a partition of $V$ into $(A, B)$ s.t. $s \in A, t \in B$, and the capacity of this cut is
$$c(A, B)=\sum_{a \in \delta^{+}(A)} c(a)$$
Clearly, for any $s$ – $t$ flow $f$ and any $s$ – $t$ cut $(A, B)$
$$\operatorname{val}(f) \leq c(A, B) \Rightarrow \max s-t \text { flow } \leq \min s-t \text { cut capacity }$$
The well-known maxflow-mincut theorem of Menger and Ford-Fulkerson states that
Theorem 1 In any directed graph, the max s-t flow value is equal to the min s-t cut capacity.

Ford and Fulkerson proved the above theorem algorithmically. Although their augmenting path algorithm is not a polynomial-time algorithm, it can be made to run in polynomial time with very slight modifications (for example, the Edmonds-Karp modification to use the shortest augmenting path). Moreover, the algorithm shows that there is an integer valued maximum flow whenever the capacities are integer valued. Thus we have

Theorem 2 In any directed graph, the max s-t flow value is equal to the min s-t cut capacity. Moreover, if $c$ is integer valued then there is an integer valued maximum flow.

This is an example of a polynomial time (good) algorithm revealing structural properties of the problem in terms of a min-max result and integrality of flows. Conversely, suppose we knew the maxflow-mincut theorem but did not know any algorithmic result. Could we gain some insight? We claim that the answer is yes. Consider the decision problem: given $G, s, t$, is the $s$ – $t$ max flow value at least some given number $k$ ? It is easy to see that this problem is in NP since one can give a feasible flow $f$ of value at least $k$ as an NP certificate ${ }^1$. However, using the maxflow-mincut theorem we can also see that it is in coNP . To show that the flow value is smaller than $k$, all we need to exhibit is a cut of capacity smaller than $k$. Therefore the min-max result shows that the problem is in NP $\cap$ coNP . Most “natural” decision problems in NP $\cap$ coNP have eventually been shown to have polynomial time algorithms (there are a few well-known exceptions). Moreover, a problem in NP $\cap$ coNP being NP -complete or coNP -complete would imply that NP = coNP , some thing that most believe is unlikely. Thus a min-max result implies that the decision version is in NP $\cap$ coNP, strong evidence for the existence of a poly-time algorithm. That does not imply that such an algorithm will come by easily. Examples include matchings in general graphs and linear programming.

# 组合优化代写

## 数学代写|组合优化代写Combinatorial optimization代考|Introduction and Motivation

(ii) 对于每个实例 $I$ 有一层 1 loperatorname{sol}(I) Isubset ISigma^ 这样 $(一) \forall s \in \operatorname{sol}(I),|s|=\operatorname{poly}(|I|)$
(b) 存在一个多时间算法，在输入 $I, \mathrm{~s}$ 正确输出是否 $s \in \operatorname{sol}(I)$ 或不 (iii) 有一个函数 val : $\Sigma^{\times} \Sigma^* \rightarrow \mathbb{Z}$ 英 石 $\operatorname{val}(I, s)$ 为每个实例分配一个整数 $I$ 和 $s \in \operatorname{sol}(I)$ 此外， val 可以通过多时间算法计算。

## 数学代写|组合优化代写Combinatorial optimization代考|Network Flow

\begin{aligned} & f: A \rightarrow \mathbb{R}^{+} \text {圣 } \ & \quad\left(\text { 一) } \sum_{a \in \delta^{-}(v)} f(a)=\sum_{a \in \delta^{+}(v)} f(a) \text { 对所有人 } v \in V-s, t\right. \ & \text { (二) } 0 \leq f(a) \leq c(a) \text { 对所有人 } a \in A \end{aligned}

$$\operatorname{val}(f)=\sum_{a \in \delta^{+}(s)} f(a)-\sum_{a \in \delta^{-}(s)} f(a)=\sum_{a \in \delta^{-}(t)} f(a)-\sum_{a \in \delta^{+}(t)} f(a)$$

$$c(A, B)=\sum_{a \in \delta^{+}(A)} c(a)$$

$$\operatorname{val}(f) \leq c(A, B) \Rightarrow \max s-t \text { flow } \leq \min s-t \text { cut capacity }$$
Menger 和 Ford-Fulkerson 著名的 maxflow-mincut 定理指出 定理 1 在任何有向图中，最大 st 流值等于最小 st 切割容量。
Ford 和 Fulkerson 通过算法证明了上述定理。尽管他们的增广路径算法不是多项式时间算法，但可以通过 非常轻微的修改使其在多项式时间内运行（例如，使用最短增广路径的 Edmonds-Karp 修改) 。此外，该 算法表明，只要容量为整数值，就有一个整数值的最大流量。因此我们有

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|组合优化代写Combinatorial optimization代考|APM6664

statistics-lab™ 为您的留学生涯保驾护航 在代写组合优化Combinatorial optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写组合优化Combinatorial optimization代写方面经验极为丰富，各种代写组合优化Combinatorial optimization相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|组合优化代写Combinatorial optimization代考|A4k(k + 1) Vertex Kernel

From the result in Sect. 4 we can devise a polynomial size kernel for CEVS. To achieve this we propose three reduction rules, prove that they are valid, and that their application gives a kernel as required.
Reduction Rule 1. Remove all isolated Cliques.
Lemma 10. Reduction Rule 1 is sound.
Proof. As no optimal solution has a final clique which bridges two connected components of a graph $G$ and an isolated clique needs no edits to make it complete, the clique will remain in all optimal solutions and as such can be removed without affecting the result.

Reduction Rule 2. Reduce all critical cliques with more than $k+1$ vertices to $k+1$ vertices.
Lemma 11. Reduction Rule 2 is sound.
Proof. As no solution clique in an optimal solution partially contains a critical clique and the cost to delete the edges incident to, or add edges to all of, or to divide the vertices of such a critical clique is greater than $k$ is too high to be allowed we only need to maintain this invariant. Thus we can remove vertices from any critical clique with more than $k+1$ vertices until there are at most $k+1$ vertices in a critical clique and this will not affect the result.

Reduction Rule 3. If there are mote than $4 k$ non-isolated critical cliques reduce the graph to a $P_3$ and set $k=0$ in this case.

## 数学代写|组合优化代写Combinatorial optimization代考|Our New Formulations

In $\left(P_2\right)$, for all $k \in \mathcal{K}$, variable $z^k$ is equal to 1 if and only if the optimal radius is greater than or equal to $D^k$. As a consequence, the following constraints are valid
$$z^k \geq z^{k+1} \quad k \in{1, \ldots, K-1} .$$
We first show that these inequalities are redundant for $\left(P_2\right)$. Let $\left(P_2^{\prime}\right)$ be the formulation obtained when constraints (4) are added to $\left(P_2\right)$ and let $v(\bar{F})$ be the optimal value of the linear relaxation of a given formulation $F$. We now prove that adding constraints (4) does not improve the quality of the linear relaxation.
Proposition 1. $v\left(\overline{P_2^{\prime}}\right)=v\left(\overline{P_2}\right)$
Proof. We show that an optimal solution $(\tilde{y}, \tilde{z})$ of the relaxation of $\left(P_2\right)$ satisfies (4). For each distance $D^k$ there exists a client $i(k)$ such that
$$\tilde{z}^k+\sum_{j: d_{i(k) j}<D^k} \tilde{y}_j=1$$
otherwise $\tilde{z}^k$ can be decreased and $(\tilde{y}, \tilde{z})$ is not optimal.

We now assume that $\tilde{z}^{k-1}<\tilde{z}^k$ for some index $k \in{2, \ldots, K}$. It follows that
$$\tilde{z}^{k-1}+\sum_{j: d_{i(k) j}<D^{k-1}} \tilde{y}j<\tilde{z}^k+\sum{j: d_{i(k) j}<D^k} \tilde{y}_j=1$$
The last equality follows from (5). Therefore, constraints (2c) for $i(k)$ and $k-1$ is violated.

# 组合优化代写

## 数学代写|组合优化代写Combinatorial optimization代考|Our New Formulations

$$z^k \geq z^{k+1} \quad k \in 1, \ldots, K-1 .$$

$$\tilde{z}^k+\sum_{j: d_{i(k) j}<D^k} \tilde{y}j=1$$ 否则 $\tilde{z}^k$ 可以减少和 $(\tilde{y}, \tilde{z})$ 不是最优的。 我们现在假设 $\tilde{z}^{k-1}<\tilde{z}^k$ 对于某些索引 $k \in 2, \ldots, K$. 它遵循 $$\tilde{z}^{k-1}+\sum{j: d_{i(k) j}<D^{k-1}} \tilde{y} j<\tilde{z}^k+\sum j: d_{i(k) j}<D^k \tilde{y}_j=1$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。