数学代写|计算复杂度理论代写Computational complexity theory代考|NP-Complete Optimization Problems

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• Statistical Inference 统计推断
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数学代写|计算复杂度理论代写Computational complexity theory代考|NP-Complete Optimization Problems

Based on the notion of polynomial-time Turing reducibility, we can see that many important combinatorial optimization problems are $N P$-hard search problems. We prove these results by first showing that the corresponding decision problems are $\leq_{m}^{P}$-complete for $N P$ and then proving that the problems of searching for the optimum solutions are $\leq_{T}^{P}$-equivalent to the corresponding decision problems. In practice, however, we often do not need the optimum solution. A nearly optimum solution is sufficient for most applications. In general, the $N P$-hardness of the optimization problem does not necessarily imply the $N P$-hardness of the approximation to the optimization problem. In this section, we demonstrate that for some $N P$-complete optimization problems, their approximation versions are also $N P$-hard and, yet, for some problems, polynomial-time approximation is achievable. These types of results are often more difficult to prove than other $N P$-completeness results. We only present some easier results and delay the more involved results until Chapter $11 .$

We first introduce a general framework to deal with the approximation problems. Very often, an optimization problem $\Pi$ has the following general structure: for each input instance $x$ to the problem $\Pi$, there are a number of solutions $y$ to $x$. For each solution $y$, we associate a value $v_{\Pi}(y)$ (or, simply, $v(y)$, if $\Pi$ is known from the context) to it. The problem $\Pi$ is to find, for the given input $x$, a solution $y$ to $x$ such that its value $v(y)$ is maximized (or minimized). For instance, we can fit the problem MAXCLIQUE into this framework as follows: an input to the problem is a graph $G$; a solution to $G$ is a clique $C$ in $G$; the value $v(C)$ of a solution $C$ is the number of its vertices; and the problem is to find, for a given graph $G$, a clique of the maximum size.

Let $r$ be a real number with $r>1$. For a maximization problem $\Pi$ with the above structure, we define its approximation version, with the approximation ratio $r$, as follows:
$r$-APProx-П: For a given input $x$, find a solution $y$ to $x$ such that $v(y) \geq v^{}(x) / r$, where $v^{}(x)=\max {v(z): z$ is a solution to $x$.
Similarly, for a minimization problem $\Pi$, its approximation version with the approximation ratio $r$ is as follows:$r$-APPROX-П: For a given input $x$, find a solution $y$ to $x$ such that $v(y) \leq r \cdot v^{}(x)$, where $v^{}(x)=\min {v(z): z$ is a solution to $x}$.

数学代写|计算复杂度理论代写Computational complexity theory代考|Nondeterministic Oracle Turing Machines

We have defined in Chapter 2 the notions of polynomial-time Turing reducibility and oracle TMs, and have seen that many optimization problems, when formulated in the search problem form, are solvable in polynomial time relative to a set in $N P$. We now extend this notion to nondeterministic oracle TMs and study problems that are solvable in nondeterministic polynomial time relative to sets in $N P$.

A nondeterministic (function-)oracle Turing machine (oracle NTM) is an NTM equipped with an additional query tape and two additional states: the query state and the answer state. The computation of an oracle NTM is similar to that of an oracle DTM, except that at each nonquery state an oracle NTM can make a nondeterministic move. We require that the query step of the computation be a deterministic move determined by the oracle. Let $M$ be an oracle NTM and $f$ an oracle function. We write $M^{f}(x)$ to denote the computation of $M$ on input $x$, using $f$ as the oracle function (note that this is a computation tree). If the oracle function is a characteristic function of a set $A$, we say $M$ is a set-oracle NTM and write $M^{A}$ to denote $M^{f}$, and write $L(M, A)$ to denote the set of strings accepted by $M^{A}$.

The time complexity of a set-oracle NTM is also defined similar to that of a set-oracle DTM. In particular, the actions from the query state to the answer state count as only one step. For any fixed oracle set $A$, we let $\operatorname{time}{M}^{A}(x)$ be the length of the shortest accepting computation path of $M^{A}(x)$ and $t{M}^{A}(n)=\max \left({n+1} \cup\left{\operatorname{time}{M}^{A}(x):|x|=n, M^{A}\right.\right.$ accepts $\left.\left.x\right}\right)$. For a set-oracle NTM $M$, we say $t{M}(n)$ is bounded by a function $g(n)$, if for all oracle sets $A, t_{M}^{A}(n) \leq g(n)$. An oracle NTM $M$ is a polynomialtime oracle $N T M$ if $t_{M}(n)$ is bounded by a polynomial function $p$. Let $A$ be a set and $\mathcal{C}$ be a complexity class. We let $N P^{A}$ denote the class of sets accepted by polynomial-time oracle NTMs relative to the oracle $A$, and let $N P^{C}$ (or, $N P(\mathcal{C})$ ) denote the class of sets accepted by polynomial-time oracle NTMs using an oracle $B \in \mathcal{C}$ (i.e., $N P^{C}=\bigcup_{B \in C} N P^{B}$ ).

数学代写|计算复杂度理论代写Computational complexity theory代考|Polynomial-Time Hierarchy

The polynomial-time hierarchy is the polynomial analog of the arithmetic hierarchy in recursion theory (Rogers, 1967). It can be defined inductively by oracle NTMs.

Definition $3.3$ For integers $n \in \mathbb{N}$, complexity classes $\Delta_{n}^{P}$, $\Sigma_{n}^{P}$, and $\Pi_{n}^{P}$ are defined as follows:
\begin{aligned} \Sigma_{0}^{P} &=\Pi_{0}^{P}=\Delta_{0}^{P}=P, \ \Sigma_{n+1}^{P} &=N P\left(\Sigma_{n}^{P}\right), \ \Pi_{n+1}^{P} &=c o-\Sigma_{n+1}^{P}, \ \Delta_{n+1}^{P} &=P\left(\Sigma_{n}^{P}\right), \quad n \geq 0 . \end{aligned}
The class $P H$ is defined to be the union of $\Sigma_{n}^{P}$ over all $n \geq 0$.
Thus, $\Sigma_{1}^{P}=N P, \Sigma_{2}^{P}=N P^{N P}, \Sigma_{3}^{P}=N P\left(N P^{N P}\right)$, and so on. It is easy to verify that these classes form a hierarchy.
Proposition 3.4 For all $k>0$,
$$\Sigma_{k}^{P} \cup \Pi_{k}^{P} \subseteq \Delta_{k+1}^{P} \subseteq \Sigma_{k+1}^{P} \cap \Pi_{k+1}^{P} \subseteq P S P A C E .$$
Proof. Note that $P^{A}=P^{\bar{A}}$, and so $\Pi_{k}^{P} \subseteq P\left(\Pi_{k}^{P}\right)=P\left(\Sigma_{k}^{P}\right)=\Delta_{k+1}^{P}$. Other inclusive relations among classes in $P H$ follow easily from the definition. Finally, the whole hierarchy $P H$ is included in $P S P A C E$ following from Proposition 3.2(b).

Based on the above proposition, we show in Figure $3.1$ the basic structure of the polynomial-time hierarchy. To further understand the structure of the polynomial-time hierarchy, we first extend Theorem $2.1$ to a characterization of the polynomial-time hierarchy in terms of the polynomiallength-bounded quantifiers.

First, we observe some closure properties of the polynomial-time hierarchy under the Boolean operations.

数学代写|计算复杂度理论代写Computational complexity theory代考|NP-Complete Optimization Problems

r-APProx-П：对于给定的输入X, 找到解决方案是至X这样在(是)≥在(X)/r, 其中 $v^{}(x)=\max {v(z): z一世s一个s○l在吨一世○n吨○X.小号一世米一世l一个rl是,F○r一个米一世n一世米一世和一个吨一世○npr○bl和米\π,一世吨s一个ppr○X一世米一个吨一世○n在和rs一世○n在一世吨H吨H和一个ppr○X一世米一个吨一世○nr一个吨一世○r一世s一个sF○ll○在s:rП−一个磷磷R○X−磷:F○r一个G一世在和n一世np在吨X,F一世nd一个s○l在吨一世○n是吨○Xs在CH吨H一个吨v(y) \leq r \cdot v^{}(x),在H和r和v ^ {} (x) = \ min {v (z): z一世s一个s○l在吨一世○n吨○x}$。

数学代写|计算复杂度理论代写Computational complexity theory代考|Nondeterministic Oracle Turing Machines

set-oracle NTM 的时间复杂度也与 set-oracle DTM 的定义类似。特别是，从查询状态到回答状态的动作仅计为一步。对于任何固定的预言机集一个，我们让时间⁡米一个(X)是最短接受计算路径的长度米一个(X)和t{M}^{A}(n)=\max \left({n+1} \cup\left{\operatorname{time}{M}^{A}(x):|x|=n, M ^{A}\right.\right.$接受$\left.\left.x\right}\right)t{M}^{A}(n)=\max \left({n+1} \cup\left{\operatorname{time}{M}^{A}(x):|x|=n, M ^{A}\right.\right.$接受$\left.\left.x\right}\right). 对于 set-oracle NTM米， 我们说吨米(n)受函数限制G(n), 如果对于所有的 oracle 集一个,吨米一个(n)≤G(n). 一个预言机 NTM米是多项式时间预言机ñ吨米如果吨米(n)以多项式函数为界p. 让一个是一个集合和C是一个复杂度类。我们让ñ磷一个表示多项式时间预言机 NTM 相对于预言机接受的集合类别一个， 然后让ñ磷C（或者，ñ磷(C)) 表示多项式时间预言机 NTM 使用预言机接受的集合类别乙∈C（IE，ñ磷C=⋃乙∈Cñ磷乙 ).

数学代写|计算复杂度理论代写Computational complexity theory代考|Polynomial-Time Hierarchy

Σ0磷=圆周率0磷=Δ0磷=磷, Σn+1磷=ñ磷(Σn磷), 圆周率n+1磷=C○−Σn+1磷, Δn+1磷=磷(Σn磷),n≥0.

Σķ磷∪圆周率ķ磷⊆Δķ+1磷⊆Σķ+1磷∩圆周率ķ+1磷⊆磷小号磷一个C和.

有限元方法代写

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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。