“计算复杂性理论”的版本间的差异

2020年11月14日 (六) 11:45的版本

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{{简介{计算问题固有难度研究}} 模板:Use mdy dates

Computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm.

计算复杂性理论关注于根据资源使用情况对计算问题进行分类，并将这些类相互关联。计算问题是由计算机解决的任务。计算问题可以通过机械地应用数学步骤来解决，例如算法。

Computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm.

计算复杂性理论Computational complexity theory的重点是根据计算问题的资源使用情况对其进行分类，并将这些类相互关联。计算问题是一个能由计算机解决的任务。计算问题是可以通过机械应用的数学步骤，例如算法来解决的。

A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do. The P versus NP problem, one of the seven Millennium Prize Problems, is dedicated to the field of computational complexity.^[1]

A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do. The P versus NP problem, one of the seven Millennium Prize Problems, is dedicated to the field of computational complexity.

如果一个问题的解决需要大量的资源，无论使用什么算法，那么这个问题本身就被认为是困难的。该理论通过引入数学的 计算模型Models of computation来研究这些问题，并量化它们的 计算复杂性，即解决这些问题所需的资源量，如时间和存储量，将这种直觉形式化。其他复杂性的度量也被使用，例如通信量(用于 通信复杂性) ，电路中的逻辑门数(用于 电路复杂性)和处理器数(用于 并行计算)。 计算复杂性理论的一个作用是确定计算机能做什么和不能做什么的实际限制。 P/NP问题，是七大 千禧年奖问题Millennium Prize Problems之一，致力于计算复杂性领域。

Closely related fields in theoretical computer science are analysis of algorithms and computability theory. A key distinction between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of resources needed by a particular algorithm to solve a problem, whereas the latter asks a more general question about all possible algorithms that could be used to solve the same problem. More precisely, computational complexity theory tries to classify problems that can or cannot be solved with appropriately restricted resources. In turn, imposing restrictions on the available resources is what distinguishes computational complexity from computability theory: the latter theory asks what kinds of problems can, in principle, be solved algorithmically.

理论计算机科学中密切相关的领域是 算法分析Analysis of algorithms和 可计算性理论Computability theory。算法分析和计算复杂性理论分析的一个关键区别在于前者致力于分析一个特定算法解决问题所需的资源量，而后者则提出了一个关于所有可能用于解决同一问题的算法的更普遍的问题。更准确地说，计算复杂性理论试图对那些能够或不能够用适当限制的资源来解决的问题进行分类。反过来，对可用资源施加限制是区分计算复杂性和可计算性理论的关键: 后者的理论提出，原则上，什么样的问题可以通过算法解决。

Computational problems计算问题

文件:TSP Deutschland 3.png

A traveling salesman tour through 14 German cities.

拇指|直立=1.5 |旅行推销员在14个德国城市旅行。 A traveling salesman tour through 14 German cities.

一个旅行推销员游览了14个德国城市。

Problem instances问题实例

A computational problem can be viewed as an infinite collection of instances together with a solution for every instance. The input string for a computational problem is referred to as a problem instance, and should not be confused with the problem itself. In computational complexity theory, a problem refers to the abstract question to be solved. In contrast, an instance of this problem is a rather concrete utterance, which can serve as the input for a decision problem. For example, consider the problem of primality testing. The instance is a number (e.g., 15) and the solution is "yes" if the number is prime and "no" otherwise (in this case, 15 is not prime and the answer is "no"). Stated another way, the instance is a particular input to the problem, and the solution is the output corresponding to the given input.

A computational problem can be viewed as an infinite collection of instances together with a solution for every instance. The input string for a computational problem is referred to as a problem instance, and should not be confused with the problem itself. In computational complexity theory, a problem refers to the abstract question to be solved. In contrast, an instance of this problem is a rather concrete utterance, which can serve as the input for a decision problem. For example, consider the problem of primality testing. The instance is a number (e.g., 15) and the solution is "yes" if the number is prime and "no" otherwise (in this case, 15 is not prime and the answer is "no"). Stated another way, the instance is a particular input to the problem, and the solution is the output corresponding to the given input.

一个 计算问题可以被看作是一个实例的无限集合和每个实例的解决方案。 计算问题的输入字符串被称为问题实例，不应与问题本身混淆。在计算复杂性理论，问题指的是要解决的抽象问题。相比之下，这个问题的实例是一个相当具体的表述，可以作为决策问题的输入。例如，考虑素性测试的问题。实例是一个数字(例如，15) ，如果数字是质数，解决方案是“是” ，否则是“否”(在这种情况下，15不是质数，答案是“否”)。换句话说，实例是问题的特定输入，解决方案是与给定输入对应的输出。

To further highlight the difference between a problem and an instance, consider the following instance of the decision version of the traveling salesman problem: Is there a route of at most 2000 kilometres passing through all of Germany's 15 largest cities? The quantitative answer to this particular problem instance is of little use for solving other instances of the problem, such as asking for a round trip through all sites in Milan whose total length is at most 10 km. For this reason, complexity theory addresses computational problems and not particular problem instances.

为了进一步突出问题和实例之间的区别，考虑以下关于旅行推销员问题的决策版本: 是否有一条最多2000公里的路线穿过德国所有15个最大的城市？这个特殊问题的定量答案对于解决这个问题的其他实例没有什么用处，例如要求在米兰的所有地点进行往返旅行，这些地点的总长度最多不超过10公里。基于这个原因， 复杂性理论解决的是 计算问题而不是特定的问题实例。

Representing problem instances问题表达实例

When considering computational problems, a problem instance is a string over an alphabet. Usually, the alphabet is taken to be the binary alphabet (i.e., the set {0,1}), and thus the strings are bitstrings. As in a real-world computer, mathematical objects other than bitstrings must be suitably encoded. For example, integers can be represented in binary notation, and graphs can be encoded directly via their adjacency matrices, or by encoding their adjacency lists in binary.

在考虑 计算问题时，一个问题实例是字母表上的字符串。通常，字母表被认为是二进制字母表(即集合{0,1}) ，因此字符串是位字符串。在现实世界的计算机中，除了位字符串以外的数学对象必须进行适当的编码。例如，整数可以用二进制表示法表示，图可以通过它们的 邻接矩阵直接进行编码，或者用二进制编码它们的 邻接列表。

Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input encoding, one tries to keep the discussion abstract enough to be independent of the choice of encoding. This can be achieved by ensuring that different representations can be transformed into each other efficiently.

尽管复杂性理论定理的一些证明通常假设输入编码的某些具体选择，但是有人试图使讨论足够抽象，使其与编码的选择无关。这可以通过确保不同的表示可以有效地相互转换来实现。

Decision problems as formal languages作为形式语言的决策问题

文件:Decision Problem.svg

A decision problem has only two possible outputs, yes or no (or alternately 1 or 0) on any input.

[[图：Decision Problem.svg|thumb | A决策问题只有两个可能的输出，任何输入上的“是”或“否”（或交替为1或0）。]]

A decision problem has only two possible outputs, yes or no (or alternately 1 or 0) on any input.

一个决策问题只有两个可能的输出，对任何输入是或否(或者交替1或0)

Decision problems are one of the central objects of study in computational complexity theory. A decision problem is a special type of computational problem whose answer is either yes or no, or alternately either 1 or 0. A decision problem can be viewed as a formal language, where the members of the language are instances whose output is yes, and the non-members are those instances whose output is no. The objective is to decide, with the aid of an algorithm, whether a given input string is a member of the formal language under consideration. If the algorithm deciding this problem returns the answer yes, the algorithm is said to accept the input string, otherwise it is said to reject the input.

Decision problems are one of the central objects of study in computational complexity theory. A decision problem is a special type of computational problem whose answer is either yes or no, or alternately either 1 or 0. A decision problem can be viewed as a formal language, where the members of the language are instances whose output is yes, and the non-members are those instances whose output is no. The objective is to decide, with the aid of an algorithm, whether a given input string is a member of the formal language under consideration. If the algorithm deciding this problem returns the answer yes, the algorithm is said to accept the input string, otherwise it is said to reject the input.

决策问题Decision problems是计算复杂性理论研究的中心内容之一。 决策问题是一种特殊类型的计算问题，其答案要么是“是”，要么是“否”，要么是“1”或“0”。 决策问题可以看作是一种形式语言，语言的成员是输出为“是”的实例，而非成员是输出为“否”的实例，其目的是借助算法来确定给定的输入字符串是否是所考虑的形式语言的成员。如果决定这个问题的算法返回的答案是“是”，则表示该算法接受输入字符串，否则表示拒绝输入。

An example of a decision problem is the following. The input is an arbitrary graph. The problem consists in deciding whether the given graph is connected or not. The formal language associated with this decision problem is then the set of all connected graphs — to obtain a precise definition of this language, one has to decide how graphs are encoded as binary strings.

决策问题的一个例子如下。输入是一个任意的图。问题是判断给定的图是否连通。与这个决策问题相关的形式语言是所有连通图的集合ーー为了获得这种语言的精确定义，必须决定图是如何编码成二进制字符串的。

Function problems函数问题

A function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem—that is, the output isn't just yes or no. Notable examples include the traveling salesman problem and the integer factorization problem.

函数问题是一个计算问题，其中每个输入都有一个输出（ 总函数），但输出比 决策问题更复杂，即输出不只是是或否。值得注意的例子包括 旅行商问题和 整数因式分解问题。

It is tempting to think that the notion of function problems is much richer than the notion of decision problems. However, this is not really the case, since function problems can be recast as decision problems. For example, the multiplication of two integers can be expressed as the set of triples (a, b, c) such that the relation a × b = c holds. Deciding whether a given triple is a member of this set corresponds to solving the problem of multiplying two numbers.

人们很容易认为 函数问题的概念比 决策问题的概念丰富得多。然而，事实并非如此，因为函数问题可以被改写为决策问题。例如，两个整数的乘法可以表示为三元组（a, b, c），这样关系a × b = c成立。决定一个给定的三元组是否是这个集合的一员相当于解决两个数相乘的问题。

Measuring the size of an instance测量实例的大小

To measure the difficulty of solving a computational problem, one may wish to see how much time the best algorithm requires to solve the problem. However, the running time may, in general, depend on the instance. In particular, larger instances will require more time to solve. Thus the time required to solve a problem (or the space required, or any measure of complexity) is calculated as a function of the size of the instance. This is usually taken to be the size of the input in bits. Complexity theory is interested in how algorithms scale with an increase in the input size. For instance, in the problem of finding whether a graph is connected, how much more time does it take to solve a problem for a graph with 2n vertices compared to the time taken for a graph with n vertices?

为了衡量解决一个计算问题的难度，人们可能希望知道最佳算法需要多少时间来解决这个问题。但是，运行时间通常取决于实例。特别是，较大的实例将需要更多的时间来解决。因此，解决一个问题所需的时间（或所需的空间，或任何复杂程度的度量）被计算为实例大小的函数。这通常被认为是以比特为单位的输入大小。复杂度理论关心的是算法如何随着输入大小的增加而伸缩。例如，在求一个图是否连通的问题中，一个2n个顶点的图比一个有n个顶点的图要多花多少时间？

If the input size is n, the time taken can be expressed as a function of n. Since the time taken on different inputs of the same size can be different, the worst-case time complexity T(n) is defined to be the maximum time taken over all inputs of size n. If T(n) is a polynomial in n, then the algorithm is said to be a polynomial time algorithm. Cobham's thesis argues that a problem can be solved with a feasible amount of resources if it admits a polynomial time algorithm.

如果输入大小为 n，所花费的时间可以表示为 n 的函数。由于同一大小的不同输入所占用的时间可能不同，最坏情况下的时间复杂度 T(n)被定义为大小为 n 的所有输入所占用的最大时间。如果 T(n)是 n 中的多项式，那么该算法称为 多项式时间Polynomial time算法。 科巴姆Cobham的论文认为，如果一个问题采用 多项式时间算法，那么它可以用可行的资源量来解决。

Machine models and complexity measures机器模型与复杂性度量

Turing machine图灵机

文件:Turing machine 2b.svg

An illustration of a Turing machine

拇指|右|图灵机器图解

An illustration of a Turing machine

图灵机的图解

A Turing machine is a mathematical model of a general computing machine. It is a theoretical device that manipulates symbols contained on a strip of tape. Turing machines are not intended as a practical computing technology, but rather as a general model of a computing machine—anything from an advanced supercomputer to a mathematician with a pencil and paper. It is believed that if a problem can be solved by an algorithm, there exists a Turing machine that solves the problem. Indeed, this is the statement of the Church–Turing thesis. Furthermore, it is known that everything that can be computed on other models of computation known to us today, such as a RAM machine, Conway's Game of Life, cellular automata or any programming language can be computed on a Turing machine. Since Turing machines are easy to analyze mathematically, and are believed to be as powerful as any other model of computation, the Turing machine is the most commonly used model in complexity theory.

图灵机Turing machine是一般计算机的数学模型。它是一种理论上的装置，可以操纵一条带子上的符号。图灵机器并不是一种实用的计算技术，而是一种计算机的通用模型，（适用于）从先进的超级计算机到有铅笔和纸的数学家。一般认为，如果一个问题可以用一个算法来解决，那么就存在一个图灵机器来解决这个问题。事实上，这正是 丘奇-图灵论题Church–Turing thesis的陈述。此外，众所周知，在我们今天已知的其他计算模型上可以计算的一切，例如 RAM机器、Conway的生命游戏、元胞自动机或任何编程语言都可以在图灵机器上进行计算。由于图灵机易于数学分析，并且被认为与任何其他计算模型一样强大， 图灵机Turing machine是复杂性理论中最常用的模型。

Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and alternating Turing machines. They are all equally powerful in principle, but when resources (such as time or space) are bounded, some of these may be more powerful than others.

许多类型的 图灵机Turing machine被用来定义复杂性类，如 确定性图灵机、概率图灵机、不确定图灵机、量子图灵机、对称图灵机和交替图灵机。它们在原则上都同样强大，但当资源(如时间或空间)被限制时，其中一些可能比其他的更强大。

A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called randomized algorithms. A non-deterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see non-deterministic algorithm.

确定性图灵机Turing machine是最基本的图灵机，它使用一组固定的规则来决定未来的动作。 概率图灵机Probabilistic Turing machine是一种具有额外随机位的确定性图灵机。做出概率决策的能力通常有助于算法更有效地解决问题。使用随机位的算法称为随机算法。 非确定型图灵机Non-deterministic Turing machine是一种确定性图灵机，具有额外的非确定性特征，它允许图灵机在给定的状态下有多种可能的未来动作。查看非确定性的一种方法是，图灵机在每个步骤中分支成许多可能的计算路径，如果它在这些分支的任何一个中解决了这个问题，就说它已经解决了这个问题。显然，这个模型并不意味着是一个物理上可实现的模型，它只是一个理论上有趣的抽象机器，它产生了特别有趣的复杂性类。例如，请参阅 非确定性算法Non-deterministic algorithm。

Other machine models其他机型

Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary.^[2] What all these models have in common is that the machines operate deterministically.

Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary. What all these models have in common is that the machines operate deterministically.

文献中提出了许多不同于标准的多带图灵机Multi-tape Turing machines 的机器模型，例如 随机存取机Random access machine。也许令人惊讶的是，这些模型中的每一个都可以转换成另一个模型，而不需要提供任何额外的计算能力。这些替代模型的时间和内存消耗可能会有所不同。所有这些模型的共同点是，这些机器都是以确定的方式运行的。

However, some computational problems are easier to analyze in terms of more unusual resources. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems.

然而，一些计算问题更容易用更多不寻常的资源来分析。例如，非确定型图灵机 是一个允许同时检查许多不同可能性的计算模型。非确定型图灵机 对我们物理上想要的计算算法没有什么影响，但是它的分支精确地捕获了我们想要分析的许多数学模型，因此非确定性时间 是分析计算问题的一个非常重要的资源。

Complexity measures复杂性度量

For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n) if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).

For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n) if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).

为了精确定义在给定的时间和空间内解决问题的意义，我们使用了确定性图灵机这样的计算模型。确定性图灵机 m 在输入 x 上所需的时间是该机在停止并输出答案(“是”或“否”)之前进行的状态转换或步骤的总数。如果 m 对每个长度 n 的输入所需的时间最多为 f (n) ，则称 m 在时间 f (n)内工作。如果存在一个在时间 f (n)内运行的图灵机，决策问题 a 可以在时间 f (n)内得到解决。由于复杂性理论对根据问题的难度对问题进行分类感兴趣，因此人们根据一些标准来定义问题集。例如，在确定性图灵机上，在时间 f (n)内可解决的问题的集合由 DTIME(f (n))表示。

Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. Complexity measures are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

可以对空间要求作类似的定义。虽然时间和空间是最著名的复杂性资源，但任何 复杂性度量Complexity measure都可以被视为计算资源。复杂性度量通常是由 布鲁姆复杂性公理Blum complexity axioms定义的。复杂性理论中使用的其他复杂性度量包括通信复杂性、电路复杂性和决策树复杂性。

The complexity of an algorithm is often expressed using big O notation.

算法的复杂性通常用大 O 符号来表示。

Best, worst and average case complexity最佳、最差和平均情况下的复杂度

文件:Sorting quicksort anim.gif

Visualization of the quicksort algorithm that has average case performance [math]\displaystyle{ \mathcal{O}(n\log n) }[/math].

[[文件：排序快速排序动画.gif|thumb |可视化快速排序算法，它具有平均情况性能[math]\displaystyle{ \mathcal{O}(n\log n) }[/math].]] Visualization of the [[quicksort algorithm that has average case performance [math]\displaystyle{ \mathcal{O}(n\log n) }[/math].]]

[[具有平均大小写性能的快速排序算法[math]\displaystyle{ \mathcal{O}(n\log n) }[/math].]]

The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different inputs of the same size. Since some inputs of size n may be faster to solve than others, we define the following complexities:

最佳、最差和平均情况复杂度是指三种不同的方法来度量相同大小的不同输入的时间复杂度(或任何其他复杂度度量)。由于一些 n 大小的输入可能比其他的更快解决，我们定义了以下复杂性:

Best-case complexity: This is the complexity of solving the problem for the best input of size n.

Best-case complexity: This is the complexity of solving the problem for the best input of size n.

最佳情况复杂性Best-case complexity: 这就是解决n大小的最佳输入问题的复杂性。

Average-case complexity: This is the complexity of solving the problem on an average. This complexity is only defined with respect to a probability distribution over the inputs. For instance, if all inputs of the same size are assumed to be equally likely to appear, the average case complexity can be defined with respect to the uniform distribution over all inputs of size n.

Average-case complexity: This is the complexity of solving the problem on an average. This complexity is only defined with respect to a probability distribution over the inputs. For instance, if all inputs of the same size are assumed to be equally likely to appear, the average case complexity can be defined with respect to the uniform distribution over all inputs of size n.

平均案例复杂度Average-case complexity: 这是解决问题的平均复杂度。这种复杂性仅仅定义在输入的概率分布上。例如，如果假定所有相同大小的输入出现的可能性相等，则可以根据大小 n 的所有输入的均匀分布来定义平均案例复杂度。

Amortized analysis: Amortized analysis considers both the costly and less costly operations together over the whole series of operations of the algorithm.

Amortized analysis: Amortized analysis considers both the costly and less costly operations together over the whole series of operations of the algorithm.

平摊分析Amortized analysis: 在算法的整个系列操作中，平摊分析同时考虑了成本较高和成本较低的操作。

Worst-case complexity: This is the complexity of solving the problem for the worst input of size n.

Worst-case complexity: This is the complexity of solving the problem for the worst input of size n.

最坏情况复杂度Worst-case complexity: 这是解决大小 n 的最坏输入问题的复杂度。

The order from cheap to costly is: Best, average (of discrete uniform distribution), amortized, worst.

从便宜到昂贵的顺序是：最佳、平均（离散均匀分布）、摊销、最差。

For example, consider the deterministic sorting algorithm quicksort. This solves the problem of sorting a list of integers that is given as the input. The worst-case is when the pivot is always the largest or smallest value in the list (so the list is never divided). In this case the algorithm takes time O(n²). If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time.

例如，考虑确定性排序算法 快速排序Quicksort。这解决了对作为输入的整数列表进行排序的问题。最坏的情况是轴总是列表中的最大值或最小值（因此列表永远不会被拆分）。在这种情况下，算法需要时间O(n²)。如果我们假设输入列表的所有可能的排列都是相同的，那么排序所花费的平均时间是O(n log n)。最好的情况发生在每次旋转将列表分成两半时，也需要O(n log n)时间。

Upper and lower bounds on the complexity of problems问题复杂性的上下界

To classify the computation time (or similar resources, such as space consumption), it is helpful to demonstrate upper and lower bounds on the maximum amount of time required by the most efficient algorithm to solve a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).

对计算时间或类似的资源进行分类时，演示最有效算法解决给定问题所需的最大时间的上下界是很有帮助的。一个算法的复杂性通常被认为是其最坏情况的复杂性，除非另有说明。分析一个特定的算法属于 算法分析Analysis of algorithms的范畴。为了给出问题时间复杂度的上界 T(n) ，只需要给出一个运行时间最多为 T(n)的特定算法。然而，证明下界要困难得多，因为下界表明了解决给定问题的所有可能的算法。“所有可能的算法”这个短语不仅包括今天已知的算法，还包括将来可能发现的任何算法。为了给出问题T(n)的下界，需要证明任何算法的时间复杂度都不能低于 T(n)。

Upper and lower bounds are usually stated using the big O notation, which hides constant factors and smaller terms. This makes the bounds independent of the specific details of the computational model used. For instance, if T(n) = 7n² + 15n + 40, in big O notation one would write T(n) = O(n²).

上下界通常使用大 O 符号来表示，它隐藏了常数因子和较小的项。这使得边界独立于使用的计算模型的具体细节。例如，如果 T(n) = 7n² + 15n + 40,在大O表示法中，人们会写T(n) = O(n²)。

Complexity classes复杂性类

Defining complexity classes定义复杂性类

A complexity class is a set of problems of related complexity. Simpler complexity classes are defined by the following factors:

A complexity class is a set of problems of related complexity. Simpler complexity classes are defined by the following factors:

复杂性类是一组相关的复杂性问题。更简单的复杂类由以下因素定义:

The type of computational problem: The most commonly used problems are decision problems. However, complexity classes can be defined based on function problems, counting problems, optimization problems, promise problems, etc.
计算问题的类型：最常用的问题是 决策问题。然而，复杂度类可以基于函数问题s、计数问题s、优化问题s、承诺问题等定义。
The model of computation: The most common model of computation is the deterministic Turing machine, but many complexity classes are based on non-deterministic Turing machines, Boolean circuits, quantum Turing machines, monotone circuits, etc.
计算模型：最常见的计算模型是 确定性图灵机，但许多复杂度类是基于 非确定性图灵机，布尔电路s，量子图灵机s，单调电路s等。
The resource (or resources) that is being bounded and the bound: These two properties are usually stated together, such as "polynomial time", "logarithmic space", "constant depth", etc.
有界的资源（或资源组）和有界资源：这两个属性通常一起表述，如“多项式时间”、“对数空间”、“常数深度”等。

Some complexity classes have complicated definitions that do not fit into this framework. Thus, a typical complexity class has a definition like the following:

一些复杂性类的复杂定义不适合这个框架。因此，一个典型的复杂性类的定义如下:

The set of decision problems solvable by a deterministic Turing machine within time f(n). (This complexity class is known as DTIME(f(n)).)

时间 f (n)内确定性图灵机可解决的决策问题集。(这个复杂性类称为 DTIME (f (n))。)

But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" 模板:Harv. This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.

But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" . This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.

但是通过一些具体的函数 f (n)来限定上面的计算时间通常会产生依赖于所选择的机器模型的复杂类。例如，语言{ xx | x 是任意二进制字符串}可以在多带图灵机上用线性时间求解，但在单带图灵机模型中需要二次时间。如果我们允许运行时间的多项式变化，Cobham-Edmonds 理论指出，”任何两个合理的和一般的计算模型的时间复杂性是多项式相关的”。这就形成了复杂度等级 P 的基础，P 是一组决策问题，可以由确定性图灵机在多项式时间内解决。相应的一组函数问题是 FP。

Important complexity classes重要的复杂性类

文件:Complexity subsets pspace.svg

A representation of the relation among complexity classes

thumb |右|表示复杂类之间的关系

A representation of the relation among complexity classes

复杂类间关系的一种表示方法

Many important complexity classes can be defined by bounding the time or space used by the algorithm. Some important complexity classes of decision problems defined in this manner are the following:

许多重要的复杂度类可以通过限定算法使用的时间或空间来定义。以这种方式定义的决策问题的一些重要的复杂性类别如下:

{ | class = “ wikitable”

Complexity class	Complexity class	复杂性类	Model of computation	Model of computation	计算模式	Resource constraint	Resource constraint	资源限制
Deterministic time			Deterministic time			确定性时间
DTIME(f(n))	DTIME(f(n))	DTIME (f (n))	Deterministic Turing machine	Deterministic Turing machine	确定性图灵机	Time O(f(n))	Time O(f(n))	时间 o (f (n))
	不会		不会		不会
P	P	p	Deterministic Turing machine	Deterministic Turing machine	确定性图灵机	Time O(poly(n))	Time O(poly(n))	时间 o (poly (n))
EXPTIME	EXPTIME	EXPTIME	Deterministic Turing machine	Deterministic Turing machine	确定性图灵机	Time O(2^poly(n))	Time O(2^poly(n))	Time o (2 < sup > poly (n) )
Non-deterministic time			Non-deterministic time			不确定时间
NTIME(f(n))	NTIME(f(n))	NTIME (f (n))	Non-deterministic Turing machine	Non-deterministic Turing machine 非确定型图灵机	Time O(f(n))	Time O(f(n))	时间 o (f (n))
	不会		不会		不会
NP	NP	NP	Non-deterministic Turing machine	Non-deterministic Turing machine 非确定型图灵机	Time O(poly(n))	Time O(poly(n))	时间 o (poly (n))
NEXPTIME	NEXPTIME	NEXPTIME	Non-deterministic Turing machine	Non-deterministic Turing machine 非确定型图灵机	Time O(2^poly(n))	Time O(2^poly(n))	Time o (2 < sup > poly (n) )

{ | class = “ wikitable”

Complexity class	Complexity class	复杂性类	Model of computation	Model of computation	计算模式	Resource constraint	Resource constraint	资源限制
Deterministic space			Deterministic space			确定性空间
DSPACE(f(n))	DSPACE(f(n))	DSPACE (f (n))	Deterministic Turing machine	Deterministic Turing machine	确定性图灵机	Space O(f(n))	Space O(f(n))	空间 o (f (n))
L	L	l	Deterministic Turing machine	Deterministic Turing machine	确定性图灵机	Space O(log n)	Space O(log n)	空格 o (log n)
PSPACE	PSPACE	PSPACE	Deterministic Turing machine	Deterministic Turing machine	确定性图灵机	Space O(poly(n))	Space O(poly(n))	空间 o (poly (n))
EXPSPACE	EXPSPACE	EXPSPACE	Deterministic Turing machine	Deterministic Turing machine	确定性图灵机	Space O(2^poly(n))	Space O(2^poly(n))	空间 o (2 < sup > poly (n) )
Non-deterministic space			Non-deterministic space			非确定性空间
NSPACE(f(n))	NSPACE(f(n))	NSPACE (f (n))	Non-deterministic Turing machine	Non-deterministic Turing machine 非确定型图灵机	Space O(f(n))	Space O(f(n))	空间 o (f (n))
NL	NL	NL	Non-deterministic Turing machine	Non-deterministic Turing machine 非确定型图灵机	Space O(log n)	Space O(log n)	空格 o (log n)
NPSPACE	NPSPACE	NPSPACE	Non-deterministic Turing machine	Non-deterministic Turing machine 非确定型图灵机	Space O(poly(n))	Space O(poly(n))	空间 o (poly (n))
NEXPSPACE	NEXPSPACE	NEXPSPACE	Non-deterministic Turing machine	Non-deterministic Turing machine 非确定型图灵机	Space O(2^poly(n))	Space O(2^poly(n))	空间 o (2 < sup > poly (n) )

|}

The logarithmic-space classes (necessarily) do not take into account the space needed to represent the problem.

对数空间类(必要地)没有考虑表示问题所需的空间。

It turns out that PSPACE = NPSPACE and EXPSPACE = NEXPSPACE by Savitch's theorem.

用 Savitch 定理证明了 PSPACE = NPSPACE 和 EXPSPACE = NEXPSPACE。

Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.

其他重要的复杂性类别包括 BPP、 ZPP 和 RP，它们使用 概率图灵机定义; AC 和 NC，它们使用 布尔电路定义; BQP 和 QMA，它们使用 量子图灵机定义。# p 是计数问题(不是决策问题)的一个重要的复杂类。类如 IP 和 AM 使用交互式证明系统定义。ALL 是所有决策问题的类别。

Hierarchy theorems层次定理

For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n²), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.

对于以这种方式定义的复杂类，我们希望证明放宽对计算时间的要求确实定义了更多的问题。特别是，虽然 DTIME (n)包含在 DTIME (n < sup > 2 )中，但是知道包含是否严格还是很有意思的。对于时间和空间要求，分别用时间和空间层次定理给出了这些问题的答案。它们之所以被称为层次定理，是因为它们通过约束各自的资源，在定义的类上诱导出一个适当的层次结构。因此，存在成对的复杂类，这样一个类可以被正确地包含在另一个类中。我们已经推导出了这种适当的集合包含，我们可以继续进行定量的陈述，说明需要多少额外的时间或空间才能增加可以解决的问题的数量。

More precisely, the time hierarchy theorem states that

更准确地说， 时间层谱定理Time hierarchy theorem声称

[math]\displaystyle{ \mathsf{DTIME}\big(f(n) \big) \subsetneq \mathsf{DTIME} \big(f(n) \sdot \log^{2}(f(n)) \big) }[/math].

大(f (n) big) subsetneq mathsf { DTIME } big (f (n) sdot log ^ {2}(f (n) big) </math > 。

The space hierarchy theorem states that

空间层谱定理Space hierarchy theorem指出

[math]\displaystyle{ \mathsf{DSPACE}\big(f(n)\big) \subsetneq \mathsf{DSPACE} \big(f(n) \sdot \log(f(n)) \big) }[/math].

The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem tells us that P is strictly contained in EXPTIME, and the space hierarchy theorem tells us that L is strictly contained in PSPACE.

时间和空间层谱定理构成了大多数复杂类分离结果的基础。例如，时间层谱定理告诉我们P严格包含在EXPTIME中，而空间层谱定理告诉我们L严格包含在PSPACE中。

Reduction简化

Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at most as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.

Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at most as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.

许多复杂性类是使用化简的概念来定义的。化简是将一个问题转化为另一个问题。它抓住了一个非正式的概念，即一个问题最多和另一个问题一样困难。例如，如果一个问题 x 可以用 y 的算法来解决，那么 x 并不比 y 困难，我们说 x 简化为 y。根据简化方法，简化有多种类型，如 Cook 简化、 Karp 简化和 Levin 简化，以及简化复杂度的界限，如多项式时间简化或对数空间简化。

The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.

最常用的简化是 多项式时间图灵归约Polynomial-time reduction。这意味着还原过程需要 多项式时间。例如，将一个整数平方的问题可以简化为两个整数相乘的问题。这意味着两个整数相乘的算法可以用来求一个整数的平方。实际上，这可以通过给乘法算法的两个输入提供相同的输入来实现。因此我们看到平方并不比乘法困难，因为平方可以简化为乘法。

This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. The notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.

This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. The notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.

这就产生了一个对于复杂类来说很难解决的问题的概念。如果C中的每一个问题都可以归结为X，那么对于一类问题C来说，问题X是困难的。因此，在C中没有问题比X更难，因为X的算法允许我们在C中解决任何问题。硬问题的概念取决于所用的约化类型。对于大于P的复杂度类，通常使用 多项式时间图灵归约。特别是对NP的难问题集就是 NP难问题集。

If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, Π₂, to another problem, Π₁, would indicate that there is no known polynomial-time solution for Π₁. This is because a polynomial-time solution to Π₁ would yield a polynomial-time solution to Π₂. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.^[3]

If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, Π₂, to another problem, Π₁, would indicate that there is no known polynomial-time solution for Π₁. This is because a polynomial-time solution to Π₁ would yield a polynomial-time solution to Π₂. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.]]

如果一个问题X在C中并且对于C来说是难问题，那么X对于C来说是完全的。这意味着X是C中最难的问题（因为许多问题可能同样困难，我们可以说X是C中最难的问题之一），因此 非确定性多项式时间复杂性类完全问题（NP完全问题）类包含了NP中最困难的问题，从某种意义上说，它们是可能不在P中的最困难的问题。因为问题P = NP未被解决，如果能够将已知的NP完全问题Π₂简化为另一个问题Π₁，则表明Π₁没有已知的多项式时间解。这是因为Π₁的多项式时间解会产生Π₂的多项式时间解。同样，由于所有NP问题都可以归结为集合，找到一个可以在多项式时间内解决的 NP完全问题意味着P = NP。]]

Important open problems重要的开放性问题

文件:Complexity classes.svg

Diagram of complexity classes provided that P ≠ NP. The existence of problems in NP outside both P and NP-complete in this case was established by Ladner.^[4]

假设P≠NP的复杂性类的拇指图。在这种情况下，P和NP完全问题的NP问题的存在性是由拉德纳Ladner证明的。

P versus NP problemP/NP问题

The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.

复杂度类P通常被视为一种数学抽象，用于建模那些允许有效算法的计算任务。这个假设被称为 科巴姆-埃德蒙兹论Cobham–Edmonds thesis。另一方面，复杂度类NP包含了许多人们想要有效解决的问题，但是没有有效的算法，例如 布尔可满足性问题、哈密顿路径问题和顶点覆盖问题。由于确定性图灵机是一种特殊的非确定性图灵机，因此很容易观察到P中的每个问题也是NP类的成员。

The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology, and the ability to find formal proofs of pure mathematics theorems. The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US$1,000,000 prize for resolving the problem.

P 是否等于 NP 的问题是理论计算机科学中最重要的开放性问题之一，因为其解的广泛意义。如果答案是肯定的，那么许多重要的问题就可以得到更有效的解决方案。这些问题包括运筹学中的各种类型的整数规划问题，物流学中的许多问题，生物学中的蛋白质结构预测，以及找到纯数学定理的正式证明的能力。美国 P/NP问题协会是由美国千禧年大奖难题克雷数学研究所提出的一个建议。为了解决这个问题，有一个100万美元的奖金。

The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.

复杂度类P通常被视为一种数学抽象，用于建模那些允许有效算法的计算任务。这个假设被称为Cobham-Edmonds论文。另一方面，复杂度类 NP包含了许多人们想要有效解决的问题，但是没有有效的算法，例如[[布尔可满足性问题]、哈密顿路径问题和顶点覆盖问题。由于确定性图灵机是一种特殊的非确定性图灵机，因此很容易观察到P中的每个问题也是NP类的成员。

The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution.^[3] If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology,^[5] and the ability to find formal proofs of pure mathematics theorems.^[6] The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US$1,000,000 prize for resolving the problem.^[7]

It was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to László Babai and Eugene Luks has run time [math]\displaystyle{ O(2^{\sqrt{n \log n}}) }[/math] for graphs with n vertices, although some recent work by Babai offers some potentially new perspectives on this.

Ladner指出，如果P≠NP，则NP中存在既不是P也不是NP完全的问题。如果图同构是NP完全的，则多项式时间层次会塌缩到第二级。由于人们普遍认为多项式族不会塌缩到任何有限水平，因此图同构不是NP完全的。尽管Babai最近的一些工作提供了一些潜在的新观点，这个问题的最佳算法，属于Lászlóbai和Eugene Luks，他们已经对于有n个顶点的图运行了时间[math]\displaystyle{ O(2^{\sqrt{n \log n}}) }[/math]。

Problems in NP not known to be in P or NP-completeNP中不知是P中或NP完全问题的问题

The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a prime factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time [math]\displaystyle{ O(e^{\left(\sqrt[3]{\frac{64}{9}}\right)\sqrt[3]{(\log n)}\sqrt[3]{(\log \log n)^2}}) }[/math] to factor an odd integer n. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.

整数因式分解问题是确定给定整数的素数因式分解的计算问题。作为一个 决策问题，它是判断输入是否有一个小于k的素数因子的问题。目前还没有有效的整数分解算法，这一事实构成了几种现代密码系统的基础，如 RSA算法。整数分解问题存在于NP和co-NP中（甚至在UP和co-UP中）。如果问题是 NP完全问题，多项式时间层次将崩溃到第一级（即NP等于co-NP）。最著名的整数因式分解算法是一般的数域筛，它需要时间[math]\displaystyle{ O(e^{\left(\sqrt[3]{\frac{64}{9}}\right)\sqrt[3]{(\log n)}\sqrt[3]{(\log \log n)^2}}) }[/math]来分解一个奇数n。但是，这个问题最著名的量子算法 Shor算法是在多项式时间内运行的。不幸的是，这一事实并不能说明非量子复杂性类的问题所在。

It was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete.^[4] Such problems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete.

Ladner指出，如果“P”≠“NP”，则“NP”中存在既不是“P”也不是“NP-complete”的问题。[[图同构问题]、离散对数问题和整数因式分解问题是被认为是NP中间问题的例子。它们是极少数的NP问题中的一部分，不知道是“P”或是“NP-complete”。

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete.^[8] If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level.^[9] Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to László Babai and Eugene Luks has run time [math]\displaystyle{ O(2^{\sqrt{n \log n}}) }[/math] for graphs with n vertices, although some recent work by Babai offers some potentially new perspectives on this.^[10]

Tractable problems are frequently identified with problems that have polynomial-time solutions (P, PTIME); this is known as the Cobham–Edmonds thesis. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. If NP is not the same as P, then NP-hard problems are also intractable in this sense.

易处理的问题经常与具有多项式时间解决方案的问题(P，PTIME)相提并论; 这就是所谓的 Cobham-Edmonds 理论。在这个意义上被认为是棘手的问题包括那些退出时间难题extime-hard。如果 NP 不等于 P，那么 NP 难问题在这个意义上也是棘手的。

The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a prime factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP^[11]). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time [math]\displaystyle{ O(e^{\left(\sqrt[3]{\frac{64}{9}}\right)\sqrt[3]{(\log n)}\sqrt[3]{(\log \log n)^2}}) }[/math]^[12] to factor an odd integer n. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.

整数因式分解问题是确定给定整数的素数因式分解的计算问题。作为一个决策问题，它是决定输入是否有一个素数因子小于“k”的问题。目前还没有有效的整数因式分解算法，这一事实构成了几种现代密码系统的基础，例如 RSA算法。整数因式分解问题出现在“NP”和“co-NP”中（甚至在UP和co-UP^[13]）。如果问题是“NP-complete”，则多项式时间层次将塌陷到其第一级（即，“NP”将等于“co-NP”）。最著名的整数因式分解算法是一般数域筛，它需要时间[math]\displaystyle{ O（e^{\left（\sqrt[3]{\frac{64}{9}\right）\sqrt[3]{（\logn）}\sqrt[3]{（\log\logn）^2}） }[/math]^[14]将奇数整数“n”作为因子。然而，这个问题最著名的量子算法，即Shor算法，确实在多项式时间内运行。不幸的是，这一事实并不能说明非量子复杂性类的问题所在。

However, this identification is inexact: a polynomial-time solution with large degree or large leading coefficient grows quickly, and may be impractical for practical size problems; conversely, an exponential-time solution that grows slowly may be practical on realistic input, or a solution that takes a long time in the worst case may take a short time in most cases or the average case, and thus still be practical. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.

然而，这种辨识是不精确的：具有大阶数或大超前系数的多项式时间解增长很快，对于实际的规模问题可能不切实际；相反，缓慢增长的指数时间解在实际输入上可能是实际的，或者在最坏情况下需要很长时间的解可能在大多数情况下需要很短的时间，因此还是一般情况下实用的。说一个问题不在P中并不意味着问题的所有大型案例都是困难的，甚至大多数都是困难的。例如，Presburger算法中的决策问题被证明不在P中，然而在大多数情况下，已经编写了在合理时间内解决该问题的算法。类似地，算法可以在小于二次的时间内解决大范围的NP完全背包问题，而SAT解算器通常处理NP完全布尔可满足性问题的大量实例。

Separations between other complexity classes其他复杂性类之间的分离

To see why exponential-time algorithms are generally unusable in practice, consider a program that makes 2ⁿ operations before halting. For small n, say 100, and assuming for the sake of example that the computer does 10¹² operations each second, the program would run for about 4 × 10¹⁰ years, which is the same order of magnitude as the age of the universe. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. However, an exponential-time algorithm that takes 1.0001ⁿ operations is practical until n gets relatively large.

要了解为什么指数时间算法在实践中通常不可用，请考虑一个在暂停之前执行2ⁿ 操作的程序。对于小 n，比如100，假设计算机每秒执行10¹² 操作，程序将运行4 × 10¹⁰ 年，这与宇宙年龄的数量级相同。即使使用速度更快的计算机，该程序也只对非常小的实例有用，从这个意义上说，问题的难解性在某种程度上与技术进步无关。然而，在 n 变得相对较大之前，使用1.0001ⁿ操作的指数时间算法是可行的。

Many known complexity classes are suspected to be unequal, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory.

许多已知的复杂性类被怀疑是不相等的，但这一点尚未得到证实。例如P'NP[[PP（complexity）|⊆''PSPACE，但是P'=PSPACE。如果“P”不等于“NP”，那么“P”也不等于“PSPACE”。由于在P和PSPACE之间有许多已知的复杂度类别，例如RP、BPP、PP、BQP、MA、PH等，所以所有这些复杂度类别都有可能塌陷为一个类别。证明这些类中的任何一个是不相等的，这将是复杂性理论的一个重大突破。

Similarly, a polynomial time algorithm is not always practical. If its running time is, say, n¹⁵, it is unreasonable to consider it efficient and it is still useless except on small instances. Indeed, in practice even n³ or n² algorithms are often impractical on realistic sizes of problems.

同样，多项式时间算法也不总是实用的。如果它的运行时间是，比如说，n¹⁵ ，那么认为它有效是不合理的，而且除了小实例之外，它仍然是无用的。事实上，在实践中，即使是 n³ 或 n² 算法对于实际问题的大小也常常是不切实际的。

Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. It is believed^[15] that NP is not equal to co-NP; however, it has not yet been proven. It is clear that if these two complexity classes are not equal then P is not equal to NP, since P=co-P. Thus if P=NP we would have co-P=co-NP whence NP=P=co-P=co-NP.

“是的，问题的答案与‘NP’的问题是相反的。据信引用错误：没有找到与</ref>对应的<ref>标签 Conversely, a problem that can be solved in practice is called a 模板:Visible anchor, literally "a problem that can be handled". The term infeasible (literally "cannot be done") is sometimes used interchangeably with intractable,^[16] though this risks confusion with a feasible solution in mathematical optimization.引用错误：没有找到与</ref>对应的<ref>标签相反，一个可以在实践中解决的问题称为“模板:Visible anchor”，字面意思是“可以处理的问题”。术语不可行（字面意思是“不能做”）有时与“棘手”互换使用，尽管这有可能与数学优化中的可行解混淆。

Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer.

在真正致力于研究算法问题的复杂性的实际研究开始之前，许多研究人员都打下了大量的基础。其中最有影响力的是阿兰 · 图灵在1936年对图灵机的定义，这被证明是一个非常健壮和灵活的计算机简化。

Zobel

Earlier papers studying problems solvable by Turing machines with specific bounded resources include on real-time computations (1962). Somewhat earlier, Boris Trakhtenbrot (1956), a pioneer in the field from the USSR, studied another specific complexity measure. As he remembers:

早期研究图灵机在特定资源限制下可解决问题的论文包括实时计算(1962)。更早些时候，Boris Trakhtenbrot (1956) ，苏联在这一领域的先驱，研究了另一个具体的复杂性度量。正如他所记得的:, Justin (2015). Writing for Computer Science

The beginning of systematic studies in computational complexity is attributed to the seminal 1965 paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard E. Stearns, which laid out the definitions of time complexity and space complexity, and proved the hierarchy theorems. In addition, in 1965 Edmonds suggested to consider a "good" algorithm to be one with running time bounded by a polynomial of the input size.

计算复杂性系统研究的开始是由于 Juris Hartmanis 和理查德·斯特恩斯在1965年发表的论文《关于算法的计算复杂性》，该论文提出了时间复杂性和空间复杂性的定义，并证明了层次定理。此外，在1965年，Edmonds 建议考虑一个”好的”算法是一个运行时间由输入大小的多项式限定的算法。. p. 132. ISBN 978-1-44716639-9. </ref>

In 1967, Manuel Blum formulated a set of axioms (now known as Blum axioms) specifying desirable properties of complexity measures on the set of computable functions and proved an important result, the so-called speed-up theorem. The field began to flourish in 1971 when the Stephen Cook and Leonid Levin proved the existence of practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.

1967年，Manuel-Blum提出了一组公理（现在称为Blum公理），规定了可计算函数集合上复杂性测度的理想性质，并证明了一个重要结果，即所谓的加速定理。这一领域在1971年开始蓬勃发展，当时斯蒂芬·库克和列奥尼德·莱文证明了实际相关问题的存在，这些问题是NP完全的。1972年，Richard Karp以其里程碑式的论文《组合问题中的可约性》（reductability In combinational Problems）将这一想法向前推进了一大步。在这篇论文中，他展示了21个不同的组合和图论问题，每一个都因其计算上的困难而臭名昭著，都是NP完全的。

Tractable problems are frequently identified with problems that have polynomial-time solutions (P, PTIME); this is known as the Cobham–Edmonds thesis. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. If NP is not the same as P, then NP-hard problems are also intractable in this sense.

可处理问题通常与具有多项式时间解的问题（P，PTIME）；这被称为Cobham–Edmonds thesis。在这个意义上，已知的棘手问题包括EXPTIME-困难的问题。如果NP与P不同，那么NP-hard问题在这个意义上也是难以解决的。

In the 1980s, much work was done on the average difficulty of solving NP-complete problems—both exactly and approximately. At that time, computational complexity theory was at its height, and it was widely believed that if a problem turned out to be NP-complete, then there was little chance of being able to work with the problem in a practical situation. However, it became increasingly clear that this is not always the case, and some authors claimed that general asymptotic results are often unimportant for typical problems arising in practice.

20世纪80年代，人们对NP完全问题的平均困难度和近似解做了大量的工作。当时，计算复杂性理论处于鼎盛时期，人们普遍认为，如果一个问题最终证明是NP完全的，那么在实际情况下处理该问题的可能性很小。然而，越来越明显的是，这种情况并不总是如此，一些作者声称，一般渐近结果往往对实际中出现的典型问题并不重要。

However, this identification is inexact: a polynomial-time solution with large degree or large leading coefficient grows quickly, and may be impractical for practical size problems; conversely, an exponential-time solution that grows slowly may be practical on realistic input, or a solution that takes a long time in the worst case may take a short time in most cases or the average case, and thus still be practical. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.

然而，这种辨识是不精确的：具有大阶数或大超前系数的多项式时间解增长很快，对于实际的规模问题可能是不切实际的；相反，缓慢增长的指数时间解在实际输入上可能是实际的，或者在最坏情况下需要很长时间的解可能需要很短的时间时间在大多数情况下还是一般情况下，因此还是实用的。说一个问题不在P中并不意味着问题的所有大型案例都是困难的，甚至大多数都是困难的。例如，Presburger algorithm中的决策问题已被证明不在P中，但在大多数情况下，已经编写了在合理时间内解决该问题的算法。类似地，算法可以在小于二次时间的大范围内解决NP完全背包问题，并且SAT solver通常处理NP完全布尔可满足性问题的大型实例。

To see why exponential-time algorithms are generally unusable in practice, consider a program that makes 2ⁿ operations before halting. For small n, say 100, and assuming for the sake of example that the computer does 10¹² operations each second, the program would run for about 4 × 10¹⁰ years, which is the same order of magnitude as the age of the universe. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. However, an exponential-time algorithm that takes 1.0001ⁿ operations is practical until n gets relatively large.

要了解为什么指数时间算法在实际中通常不可用，请考虑一个在停止之前进行2ⁿ操作的程序。对于小的“n”，比如100，假设计算机每秒进行10¹² 操作，程序将运行大约4 × 10¹⁰年，与宇宙年龄的数量级相同。即使有一台速度快得多的计算机，该程序也只对非常小的实例有用，从这个意义上说，一个问题的棘手程度在某种程度上与技术进步无关。然而，在“n”变得相对较大之前，需要1.0001ⁿ 运算的指数时间算法是可行的。

Similarly, a polynomial time algorithm is not always practical. If its running time is, say, n¹⁵, it is unreasonable to consider it efficient and it is still useless except on small instances. Indeed, in practice even n³ or n² algorithms are often impractical on realistic sizes of problems.

类似地，多项式时间算法并不总是实用的。如果它的运行时间是，比如说“n”¹⁵，那么认为它是有效的是不合理的，而且除了在小实例上，它仍然是无用的。实际上，在实践中，即使是n³ 或 n² 算法对于实际大小的问题往往是不切实际的。

Continuous complexity theory连续复杂性理论

Continuous complexity theory can refer to complexity theory of problems that involve continuous functions that are approximated by discretizations, as studied in numerical analysis. One approach to complexity theory of numerical analysis^[17] is information based complexity. 连续复杂性理论可以指问题的复杂性理论，这些问题涉及通过离散化近似的连续函数，如数值分析中所研究的那样。数值分析复杂性理论的一种方法是基于信息的复杂性。

Continuous complexity theory can also refer to complexity theory of the use of analog computation, which uses continuous dynamical systems and differential equations.^[18] Control theory can be considered a form of computation and differential equations are used in the modelling of continuous-time and hybrid discrete-continuous-time systems.^[19]

连续复杂性理论也可以参考复杂性理论中使用的模拟计算，它使用连续的动力系统和微分方程s。控制理论可以看作是一种计算形式，微分方程用于连续时间和混合离散连续时间系统的建模。

History历史

An early example of algorithm complexity analysis is the running time analysis of the Euclidean algorithm done by Gabriel Lamé in 1844.

算法复杂性分析的一个早期例子是1844年Gabriel Lamé对Euclidean algorithm的运行时间分析。

Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer.

在明确致力于算法问题复杂性的实际研究开始之前，各种研究人员已经奠定了大量的基础。其中最具影响力的是1936年Alan Turing对图灵机器的定义，这是一种非常健壮和灵活的计算机简化。

The beginning of systematic studies in computational complexity is attributed to the seminal 1965 paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard E. Stearns, which laid out the definitions of time complexity and space complexity, and proved the hierarchy theorems.^[20] In addition, in 1965 Edmonds suggested to consider a "good" algorithm to be one with running time bounded by a polynomial of the input size.^[21]

系统地研究计算复杂性的开端，要归功于1965年Juris Hartmanis和Richard E.Stearns的开创性论文《算法的计算复杂性》，提出了时间复杂度和空间复杂度的定义，并证明了层次定理。

Earlier papers studying problems solvable by Turing machines with specific bounded resources include^[20] John Myhill's definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets (1961), as well as Hisao Yamada's paper^[22] on real-time computations (1962). Somewhat earlier, Boris Trakhtenbrot (1956), a pioneer in the field from the USSR, studied another specific complexity measure.^[23] As he remembers:

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In 1967, Manuel Blum formulated a set of axioms (now known as Blum axioms) specifying desirable properties of complexity measures on the set of computable functions and proved an important result, the so-called speed-up theorem. The field began to flourish in 1971 when the Stephen Cook and Leonid Levin proved the existence of practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.^[24]

1967年，Manuel Blum提出了一组公理s（现在称为Blum axioms），规定了可计算函数集合上复杂性测度的理想性质，并证明了一个重要结果，即所谓的加速定理。这个领域在1971年开始蓬勃发展，当时Stephen Cook和Leonid Levin[[Cook–Levin定理|证明了NP complete实际相关问题的存在。1972年，Richard Karp以其里程碑式的论文《组合问题中的可还原性》（Reductibility In Combinational Problems）将这一想法向前推进了一大步，在这篇论文中，他展示了21个不同的组合和图论问题，每一个都因其计算难处理而臭名昭著，是NP完全的。引用错误：没有找到与</ref>对应的<ref>标签

20世纪80年代，人们对NP完全问题的平均困难度和近似解做了大量的工作。当时，计算复杂性理论处于鼎盛时期，人们普遍认为，如果一个问题最终证明是NP完全的，那么在实际情况下处理该问题的可能性很小。然而，越来越清楚的是，{cn | reason=passive，no source | date=2019年8月}的情况并不总是如此，一些作者声称，对于实践中出现的典型问题，一般渐近结果往往不重要。

Works on complexity复杂性的研究

| location=Berlin, New York

| 地点: 柏林，纽约

Wuppuluri, Shyam; Doria, Francisco A., eds. (2020), Unravelling Complexity: The Life and Work of Gregory Chaitin, World Scientific, doi:10.1142/11270, ISBN 978-981-12-0006-9 {{citation}}: External link in |ref= (help)

| year=1999

1999年

| isbn=9780387948836

9780387948836

References参考文献

| series=Monographs in Computer Science

系列 = 计算机科学专著

Citations 引用

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↑ "P vs NP Problem | Clay Mathematics Institute". www.claymath.org (in English).
↑ See Arora & Barak 2009, Chapter 1: The computational model and why it doesn't matter
↑ ^3.0 ^3.1 See Sipser 2006, Chapter 7: Time complexity
↑ ^4.0 ^4.1 Ladner, Richard E. (1975), "On the structure of polynomial time reducibility", Journal of the ACM, 22 (1): 151–171, doi:10.1145/321864.321877.
↑ Berger, Bonnie A.; Leighton, T (1998), "Protein folding in the hydrophobic-hydrophilic (HP) model is NP-complete", Journal of Computational Biology, 5 (1): 27–40, CiteSeerX 10.1.1.139.5547, doi:10.1089/cmb.1998.5.27, PMID 9541869.
↑ Cook, Stephen (April 2000), The P versus NP Problem (PDF), Clay Mathematics Institute, archived from the original (PDF) on December 12, 2010, retrieved October 18, 2006.
↑ Jaffe, Arthur M. (2006), "The Millennium Grand Challenge in Mathematics" (PDF), Notices of the AMS, 53 (6), retrieved October 18, 2006.
↑
{{Citation 图同构问题是确定两个有限图是否为同构的计算问题。复杂度理论中一个重要的未解决的问题是图的同构问题是在“P”、“NP-完全”还是NP-中间。答案不得而知，但人们相信这个问题至少不是NP完全的。 Many known complexity classes are suspected to be unequal, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory. 许多已知的复杂类被怀疑是不平等的，但是这还没有被证明。例如 p something NP something PP something PSPACE，但 p = PSPACE 是可能的。如果 p 不等于 NP，那么 p 也不等于 PSPACE。由于 p 和 PSPACE 之间有许多已知的复杂类，如 RP、 BPP、 PP、 BQP、 MA、 PH 等，所有这些复杂类都可能坍缩成一个类。证明这些等级中的任何一个都是不平等的，将是复杂性理论的一个重大突破。 | first1 = Vikraman | last1 = Arvind Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. It is believed that NP is not equal to co-NP; however, it has not yet been proven. It is clear that if these two complexity classes are not equal then P is not equal to NP, since P=co-P. Thus if P=NP we would have co-P=co-NP whence NP=P=co-P=co-NP. 同样地，co-NP 也是包含补语问题的类(例如:。问题的是/否回答颠倒)的 NP 问题。人们普遍认为，NP 并不等同于共 NP，然而，这一观点尚未得到证实。很明显，如果这两个复杂度类不相等，那么 p 就不等于 NP，因为 p = co-P。因此，如果 p = NP，我们将得到 co-P = co-NP，其中 NP = p = co-P = co-NP。 | first2 = Piyush P. | last2 = Kurur Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes. 同样，l (所有可以在对数空间中解决的问题的集合)是否严格包含在 p 中或等于 p。同样，在两者之间存在着许多复杂类，如 NL 和 NC，它们是不同的还是相等的类。 | title = Graph isomorphism is in SPP | journal = Information and Computation It is suspected that P and BPP are equal. However, it is currently open if BPP = NEXP. 人们怀疑 P 和 BPP 是相等的。但是，如果 BPP = NEXP，它目前是开放的。 | volume = 204 | issue = 5
Intractability难处理性

= = 棘手 = = = < ! -- 本节链接于 minimax，棘手，棘手 -- >
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A problem that can be solved in theory (e.g. given large but finite resources, especially time), but for which in practice any solution takes too many resources to be useful, is known as an . Conversely, a problem that can be solved in practice is called a , literally "a problem that can be handled". The term infeasible (literally "cannot be done") is sometimes used interchangeably with intractable, though this risks confusion with a feasible solution in mathematical optimization.
在理论上可以解决的问题。但是在实践中，任何解决方案都需要太多的资源才能发挥作用。相反，一个在实践中可以解决的问题被称为“一个可以处理的问题” ，字面意思是“一个可以处理的问题”。Infeasible (字面上的意思是“不可能完成”)这个词有时候可以和棘手的问题互换使用，尽管这有可能在21最优化被混淆为一个可行的解决方案。
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↑ Schöning, Uwe (1987). Graph isomorphism is in the low hierarchy. Lecture Notes in Computer Science. 1987. pp. 114–124. doi:10.1007/bfb0039599. ISBN 978-3-540-17219-2.
↑ Babai, László (2016). "Graph Isomorphism in Quasipolynomial Time". arXiv:1512.03547 [cs.DS].
↑ Fortnow, Lance (September 13, 2002). "Computational Complexity Blog: Factoring". weblog.fortnow.com.
↑ Wolfram MathWorld: Number Field Sieve
↑ {cite web | first=Lance | last=Fortnow | authorlink=Lance Fortnow | title=计算复杂性博客：Factoring | date=2002-09-13 | url=http://weblog.fortnow.com/2002/09/complexity-class-of-week-factoring.html%7C网站=博客.fortnow.com}}
↑ Wolfram MathWorld:[1]
↑ Boaz Barak's course on Computational Complexity Lecture 2
↑ Meurant, Gerard (2014). Algorithms and Complexity. p. p. 4. ISBN 978-0-08093391-7.
↑ Smale, Steve (1997). "Complexity Theory and Numerical Analysis". Acta Numerica. Cambridge Univ Press. 6: 523–551. Bibcode:1997AcNum...6..523S. doi:10.1017/s0962492900002774. 模板:Citeseerx.
↑ Babai, László; Campagnolo, Manuel (2009). "A Survey on Continuous Time Computations". arXiv:0907.3117 [cs.CC].
↑ Tomlin, Claire J.; Mitchell, Ian; Bayen, Alexandre M.; Oishi, Meeko (July 2003). "Computational Techniques for the Verification of Hybrid Systems". Proceedings of the IEEE. 91 (7): 986–1001. doi:10.1109/jproc.2003.814621. 模板:Citeseerx.
↑ ^20.0 ^20.1 Fortnow & Homer (2003)
↑ Richard M. Karp, "Combinatorics, Complexity, and Randomness", 1985 Turing Award Lecture
↑ Yamada, H. (1962). "Real-Time Computation and Recursive Functions Not Real-Time Computable". IEEE Transactions on Electronic Computers. EC-11 (6): 753–760. doi:10.1109/TEC.1962.5219459.
↑ Trakhtenbrot, B.A.: Signalizing functions and tabular operators. Uchionnye Zapiski 早期研究图灵机器在特定有界资源下可解问题的论文包括Yamada, H. "实时计算和递归函数不可实时计算". IEEE电子计算机事务. doi:10.1109/TEC.1962.5219459. {{cite journal}}: Unknown parameter |卷= ignored (help); Unknown parameter |年= ignored (help); Unknown parameter |问题= ignored (help); Unknown parameter |页= ignored (help) Penzenskogo Pedinstituta (Transactions of the Penza Pedagogoical Institute) 4, 75–87 (1956) (in Russian)
↑ Richard M. Karp (1972), "Reducibility Among Combinatorial Problems" (PDF), in R. E. Miller; J. W. Thatcher (eds.), Complexity of Computer Computations, New York: Plenum, pp. 85–103

| last=Du

| last = Du

Textbooks教材

| first=Ding-Zhu

| 第一 = 鼎柱

Arora, Sanjeev; Barak, Boaz (2000

2000年), Computational Complexity: A Modern Approach, Cambridge University Press, ISBN 978-0-471-34506-0 {{citation}}: Check date values in: |year= (help); More than one of |author2= and |last2= specified (help); line feed character in |year= at position 5 (help)

}}

| year=2009

| isbn=978-0-521-42426-4

| zbl=1193.68112

| last=Goldreich

| last = Goldreich

}}

| first=Oded

第一个 = Oded

Downey, Rod; Fellows, Michael (2008

2008年), [http://www.wisdom.weizmann.ac.il/~oded/cc-book.html

Http://www.wisdom.weizmann.ac.il/~oded/cc-book.html 计算复杂性: 一个概念性的视角], Cambridge University Press

剑桥大学出版社 {{citation}}: Check |url= value (help); Check date values in: |year= (help); line feed character in |publisher= at position 27 (help); line feed character in |url= at position 52 (help); line feed character in |year= at position 5 (help)

}}

| title=Parameterized complexity

| url=https://www.springer.com/sgw/cda/frontpage/0,11855,5-0-22-1519914-0,00.html

| editor1-last=van Leeuwen

| publisher=Springer-Verlag

| editor1-first=Jan

1-first = Jan

| location=Berlin, New York

| editor1-link = Jan van Leeuwen

| year=1999

| title=Handbook of theoretical computer science (vol. A): algorithms and complexity

| title = 理论计算机科学手册。A)算法和复杂性

| isbn=9780387948836

| publisher=MIT Press

| publisher = MIT Press

| series=Monographs in Computer Science

| isbn=978-0-444-88071-0

| isbn = 978-0-444-88071-0

}}

| year=1990

1990年

{{citation}}: Empty citation (help)

}}

| last=Du

| first=Ding-Zhu

| last = Papadimitriou

| last = Papadimitriou

| author2=Ko, Ker-I

| first = Christos

第一季，克里斯托

| title=Theory of Computational Complexity

| authorlink = Christos Papadimitriou

作者/链接 = 赫里斯托斯·帕帕季米特里乌

| publisher=John Wiley & Sons

| title = Computational Complexity

| title = 计算复杂性

| year=2000

| edition = 1st

1st

| isbn=978-0-471-34506-0

| year = 1994

1994年

}}

| publisher = Addison Wesley

艾迪生 · 韦斯利

模板:Garey-Johnson

| isbn = 978-0-201-53082-7

| isbn = 978-0-201-53082-7

{{citation}}: Empty citation (help)

}}

| last=Goldreich

| first=Oded

|last=Sipser

最后 = Sipser

| authorlink=Oded Goldreich

|first=Michael

迈克尔

| url = http://www.wisdom.weizmann.ac.il/~oded/cc-book.html

|authorlink=Michael Sipser

| authorlink = Michael Sipser

| title = Computational Complexity: A Conceptual Perspective

|title=Introduction to the Theory of Computation

| title = 美国计算理论学会简介

| publisher = Cambridge University Press

|edition=2nd

2nd

| year = 2008

|year=2006

2006年

}}

|publisher=Thomson Course Technology

| publisher = Thomson Course Technology

van Leeuwen, Jan (ed.), USA, ISBN 978-0-534-95097-2 {{citation}}: Missing or empty |title= (help)

}}