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

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• 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

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## MATLAB代写

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