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Combinatorial optimization <!-- synonymous or subfield?: discrete optimization--> is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not tractable. It operates on the domain of those optimization problems in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution.  Typical problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem.

Combinatorial optimization <!-- synonymous or subfield?: discrete optimization--> is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not tractable. It operates on the domain of those optimization problems in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution.  Typical problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem.

组合优化是一个主题，包括从一个有限的对象集合中寻找一个最佳对象。<ref>{{harvnb|Schrijver|2003|p=1}}.</ref>在许多这样的问题中，穷举搜索是不易处理的。它是在可行解集是离散的或可以化为离散的优化问题的域上进行运算的，其目标是找到最优解。典型的问题是'''<font color="#FF8000">旅行商问题 Traveling Salesman Problem </font>'''(“ TSP”)、最小生成树问题(“ MST”)和'''<font color="#FF8000">背包问题 Knapsack Problem </font>'''。

组合优化主要是从一个有限的对象集合中寻找一个最佳对象。<ref>{{harvnb|Schrijver|2003|p=1}}.</ref>在许多这样的问题中，'''<font color="#FF8000">穷举搜索 exhaustive search </font>'''是不易处理的。如果这些优化问题可行解集是离散的，或者可行解集可以化为离散的，那么可以在问题范围内进行运算，其目标是找到最优解。典型的问题是'''<font color="#FF8000">旅行商问题 Traveling Salesman Problem </font>'''(“ TSP”)、最小生成树问题(“ MST”)和'''<font color="#FF8000">背包问题 Knapsack Problem </font>'''。

 

* polynomial-time exactly solvable special cases of the problem at hand (e.g. see [[fixed-parameter tractable]])

* polynomial-time exactly solvable special cases of the problem at hand (e.g. see [[fixed-parameter tractable]])

多项式时间可精确解决的特殊问题（例如，见 固定易处理的参数）

多项式时间可精确解决的特殊问题（例如，见 固定参数可解）

* algorithms that perform well on "random" instances (e.g. for [[Traveling salesman problem#TSP path length for random pointset in a square|TSP]])

* algorithms that perform well on "random" instances (e.g. for [[Traveling salesman problem#TSP path length for random pointset in a square|TSP]])

在“随机”实例上表现良好的算法（例如， TSP）

在“随机”实例上表现良好的算法（例如， TSP）

Formally, a combinatorial optimization problem <math>A</math> is a quadruple <math>(I, f, m, g)</math>, where

Formally, a combinatorial optimization problem <math>A</math> is a quadruple <math>(I, f, m, g)</math>, where

从形式上来说，一个组合优化问题<math>A</math>是一个关于四变量<math>(I，f，m，g)</math>的问题 ：

从形式上来说，一个组合优化问题<math>A</math>是涉及四个变量<math>(I，f，m，g)</math>的问题 ：

* <math>I</math> is a [[Set (mathematics)|set]] of instances;

* <math>I</math> is a [[Set (mathematics)|set]] of instances;

<math>I</math>是实例的数学中的集合；

<math>I</math>是实例中的数学集合；

* given an instance <math>x \in I</math>, <math>f(x)</math> is the finite set of feasible solutions;

* given an instance <math>x \in I</math>, <math>f(x)</math> is the finite set of feasible solutions;

给定<math>I</math>中的一个实例，<math>f(x)</math>是可行解的有限集合；

给定<math>I</math>中的一个实例，<math>f(x)</math>是可行解的有限集合；

* given an instance <math>x</math> and a feasible solution <math>y</math> of <math>x</math>, <math>m(x, y)</math> denotes the [[Measure (mathematics)|measure]] of <math>y</math>, which is usually a [[Positive (mathematics)|positive]] [[Real number|real]].

* given an instance <math>x</math> and a feasible solution <math>y</math> of <math>x</math>, <math>m(x, y)</math> denotes the [[Measure (mathematics)|measure]] of <math>y</math>, which is usually a [[Positive (mathematics)|positive]] [[Real number|real]].

给定一个实例<math>x</math>和一个可行解<math>y</math>，<math>m(x,y)</math>表示<math>y</math>的<font color="#32cd32"> 度量，其通常是一个正实数。</font>

给定一个实例<math>x</math>和其对应的可行解<math>y</math>，<math>m(x,y)</math>表示<math>y</math>的<font color="#32cd32"> 测度，其中，y通常是正实数。</font>

* <math>g</math> is the goal function, and is either <math>\min</math> or <math>\max</math>.

* <math>g</math> is the goal function, and is either <math>\min</math> or <math>\max</math>.

<math>g</math>是目标函数，可以是求最小值也可以是最大值。

<math>g</math>是目标函数，可以是最小值也可以是最大值。

The goal is then to find for some instance <math>x</math> an optimal solution, that is, a feasible solution <math>y</math>  

The goal is then to find for some instance <math>x</math> an optimal solution, that is, a feasible solution <math>y</math>  

然后，我们的目标是找到一个最优解<math>x</math>，也就是一个可行的解<math>y</math>。

然后，我们的目标是找到实例<math>x</math>的一个最优解，也就是可行解<math>y</math>。

In the field of approximation algorithms, algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.

In the field of approximation algorithms, algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.

在'''<font color="#ff8000"> 近似算法领域 approximation algorithms</font>'''，算法被设计来寻找困难问题的近似最优解。因此，通常的决策版本对问题的定义不够充分，因为它只指定了可接受的解决办法。尽管我们可以引入合适的决策问题，使这个问题更自然地被描述为一个最优化问题。<ref name="Ausiello03">{{citation|last1=Ausiello|first1=Giorgio|title=Complexity and Approximation|year=2003|edition=Corrected|publisher=Springer|isbn=978-3-540-65431-5|display-authors=etal}}</ref>

在'''<font color="#ff8000"> 近似算法approximation algorithms</font>'''领域，算法被设计来寻找困难问题的近似最优解。因此，通常的决策版本对问题的定义不够充分，因为它只指定了可接受的解决办法。尽管我们可以引入合适的决策问题，使这个问题更自然地被描述为一个最优化问题。<ref name="Ausiello03">{{citation|last1=Ausiello|first1=Giorgio|title=Complexity and Approximation|year=2003|edition=Corrected|publisher=Springer|isbn=978-3-540-65431-5|display-authors=etal}}</ref>

 

An ''NP-optimization problem'' (NPO) is a combinatorial optimization problem with the following additional conditions.<ref name="Hromkovic02">{{citation|last1=Hromkovic|first1=Juraj|title=Algorithmics for Hard Problems|year=2002|series=Texts in Theoretical Computer Science|edition=2nd|publisher=Springer|isbn=978-3-540-44134-2}}</ref> Note that the below referred [[Polynomial|polynomials]] are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

An ''NP-optimization problem'' (NPO) is a combinatorial optimization problem with the following additional conditions.<ref name="Hromkovic02">{{citation|last1=Hromkovic|first1=Juraj|title=Algorithmics for Hard Problems|year=2002|series=Texts in Theoretical Computer Science|edition=2nd|publisher=Springer|isbn=978-3-540-44134-2}}</ref> Note that the below referred [[Polynomial|polynomials]] are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

An NP-optimization problem (NPO) is a combinatorial optimization problem with the following additional conditions. Note that the below referred polynomials are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

An NP-optimization problem (NPO) is a combinatorial optimization problem with the following additional conditions. Note that the below referred polynomials are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

<font color="#32cd32"> NP优化问题（NPO）是一个具有以下附加条件的组合优化问题。<ref name="Hromkovic02">{{citation|last1=Hromkovic|first1=Juraj|title=Algorithmics for Hard Problems|year=2002|series=Texts in Theoretical Computer Science|edition=2nd|publisher=Springer|isbn=978-3-540-44134-2}}</ref>注意，下面提到的多项式是相应函数输入大小的函数，而不是某些隐式输入实例集大小的函数。

NP优化问题（NPO）是一个具有以下附加条件的组合优化问题。<ref name="Hromkovic02">{{citation|last1=Hromkovic|first1=Juraj|title=Algorithmics for Hard Problems|year=2002|series=Texts in Theoretical Computer Science|edition=2nd|publisher=Springer|isbn=978-3-540-44134-2}}</ref>注意，下面提到的多项式是相应函数输入大小的函数，而不是某些隐式输入实例集大小的函数。

* the size of every feasible solution <math>y\in f(x)</math> is polynomially [[Bounded set|bounded]] in the size of the given instance <math>x</math>,

* the size of every feasible solution <math>y\in f(x)</math> is polynomially [[Bounded set|bounded]] in the size of the given instance <math>x</math>,

在<math>f(x)</math>中，对给定实例的大小<math>x</math>中，每个可行解的大小都是多项式有界的，

每个可行解<math>y\in f(x)</math>的大小都是由给定实例<math>x</math>的大小多项式约束的，

* the languages <math>\{\,x\,\mid\, x \in I \,\}</math> and <math>\{\,(x,y)\, \mid\, y \in f(x) \,\}</math> can be [[Decidable language|recognized]] in [[polynomial time]], and

* the languages <math>\{\,x\,\mid\, x \in I \,\}</math> and <math>\{\,(x,y)\, \mid\, y \in f(x) \,\}</math> can be [[Decidable language|recognized]] in [[polynomial time]], and

语言<math>\{\,x\,\mid\,x\ in I\,\}</math>和<math>f(x)\,\}</math>中的<math>\{\,(x,y)\,\mid\,y\</math>在多项式时间内可以可判定语言/识别，并且，</font>

语言<math>\{\,x\,\mid\, x \in I \,\}</math> and <math>\{\,(x,y)\, \mid\, y \in f(x) \,\}</math>在多项式时间内可以可判定语言/识别，并且，

 

 

 

* <math>m</math> is [[Polynomial time|polynomial-time computable]].

* <math>m</math> is [[Polynomial time|polynomial-time computable]].

<math>m</math>是可计算的多项式时间。

<math>m</math>是可计算的多项式时间。

This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-complete. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions. An example of such a reduction would be the L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.

This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-complete. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions. An example of such a reduction would be the L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.

这意味着相应的决策问题在NP中。在计算机科学中，有趣的优化问题通常具有上述性质，因此是NPO问题。如果存在一种在多项式时间内找到最优解的算法，则该问题又称为'''<font color="#FF8000">P-优化（PO）问题 P-optimization problem </font>'''。通常，在处理NPO类问题时，人们对决策版本为NP完全的优化问题感兴趣。请注意，硬度关系总是与某些降低有关。由于近似算法和计算优化问题之间的联系，在某些方面保持近似性的缩减比通常的'''<font color="#FF8000">图灵和卡普约化 Turing and Karp Reductions </font>'''更为可取。这种减少的一个例子就是'''<font color="#FF8000">L-约化 L-reduction </font>'''。因此，具有NP完全决策版本的优化问题不一定称为NPO完全问题。<ref name="Kann92">{{citation|last1=Kann|first1=Viggo|title=On the Approximability of NP-complete Optimization Problems|year=1992|publisher=Royal Institute of Technology, Sweden|isbn=91-7170-082-X}}</ref>

这意味着相应的决策问题在NP中。在计算机科学中，有趣的优化问题通常具有上述性质，因此是NPO问题。如果存在一种在多项式时间内找到最优解的算法，则该问题又称为'''<font color="#FF8000">P-优化（PO）问题 P-optimization problem </font>'''。通常，在处理NPO类问题时，人们对决策版本为NP完全的优化问题感兴趣。请注意，硬度关系总是与某些降低有关。由于近似算法和计算优化问题之间的联系，在某些方面保持近似性的缩减比一般的'''<font color="#FF8000">图灵和卡普规约 Turing and Karp Reductions </font>'''更为可取。这种规约的一个例子就是'''<font color="#FF8000">L-规约 L-reduction </font>'''。因此，具有NP完全决策版本的优化问题不一定称为NPO完全问题。<ref name="Kann92">{{citation|last1=Kann|first1=Viggo|title=On the Approximability of NP-complete Optimization Problems|year=1992|publisher=Royal Institute of Technology, Sweden|isbn=91-7170-082-X}}</ref>

* ''NPO(I)'': Equals [[FPTAS]]. Contains the [[Knapsack problem]].

* ''NPO(I)'': Equals [[FPTAS]]. Contains the [[Knapsack problem]].

''NPO(I)''：等价于[[FPTAS]]。包含背包问题。

''NPO(I)''：等价于'''<font color="#FF8000">完全多项式时间近似方案 Fully Polynomial-time approximation scheme | PTAS </font>'''。包含背包问题。

* ''NPO(II)'': Equals [[Polynomial-time approximation scheme|PTAS]]. Contains the [[Makespan]] scheduling problem.

* ''NPO(II)'': Equals [[Polynomial-time approximation scheme|PTAS]]. Contains the [[Makespan]] scheduling problem.

''NPO(II)''：等价于 多项式时间近似方案| PTAS 。包含分批调度问题。

''NPO(II)''：等价于'''<font color="#FF8000">多项式时间近似方案 Polynomial-time approximation scheme | PTAS </font>''' 。包含分批调度问题。

* ''NPO(III)'': :The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at most ''c'' times the optimal cost (for minimization problems) or a cost at least <math>1/c</math> of the optimal cost (for maximization problems). In [[Juraj Hromkovič|Hromkovič]]'s book, excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains [[MAX-SAT]] and metric [[Travelling salesman problem|TSP]].

* ''NPO(III)'': :The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at most ''c'' times the optimal cost (for minimization problems) or a cost at least <math>1/c</math> of the optimal cost (for maximization problems). In [[Juraj Hromkovič|Hromkovič]]'s book, excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains [[MAX-SAT]] and metric [[Travelling salesman problem|TSP]].

''NPO(III)''：具有多项式时间算法的NPO问题类，其计算的解的成本最多为最优成本的“c”倍（对于最小化问题），或成本至少为最优成本的<math>1/c</math>（对于最大化问题）。在尤拉·赫罗姆科维奇 Juraj Hromkovic的书中，除了P=NP之外，所有的NPO(II)问题都被排除在这个类之外。如果没有排除，则等于APX。包含MAX-SAT和标准的旅行商问题| TSP。

''NPO(III)''：具有多项式时间算法的NPO问题类，其计算的解的成本最多为最优成本的“c”倍（对于最小化问题），或成本至少为最优成本的<math>1/c</math>（对于最大化问题）。在尤拉·赫罗姆科维奇 Juraj Hromkovic 的书中，除了P=NP之外，所有的NPO(II)问题都被排除在这个类之外。如果没有排除，则等于APX（approximable）。包含'''<font color="#FF8000">最大可满足性问题 MAX-SAT </font>'''和标准的旅行商问题| TSP。

* ''NPO(IV)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio that is polynomial in a logarithm of the size of the input. In Hromkovic's book, all NPO(III)-problems are excluded from this class unless P=NP. Contains the [[set cover]] problem.

* ''NPO(IV)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio that is polynomial in a logarithm of the size of the input. In Hromkovic's book, all NPO(III)-problems are excluded from this class unless P=NP. Contains the [[set cover]] problem.

''NPO(IV)''：多项式时间算法的一类NPO问题，以比率为输入大小的对数多项式来逼近最优解。在Hromkovic的书中，除非P=NP，否则所有的NPO(III)-问题都不属于此类。包含集合覆盖问题。

''NPO(IV)''：多项式时间算法的一类NPO问题，以比率为输入大小的对数多项式来逼近最优解。在Hromkovic的书中，除非P=NP，否则所有的NPO(III)-问题都不属于此类。包含集合覆盖问题。

''NPO(V)''：多项式时间算法的一类NPO问题，以某个函数限定的比率来逼近最优解。在Hromkovic的书中，除非P=NP，否则所有NPO(IV)-问题都不属于这类问题。包含旅行商问题| TSP和集团问题|最大集团问题。

''NPO(V)''：多项式时间算法的一类NPO问题，以某个函数限定的比率来逼近最优解。在Hromkovic的书中，除非P=NP，否则所有NPO(IV)-问题都不属于这类问题。包含旅行商问题| TSP和集团问题|最大集团问题。

如果对于每个实例<math>x</math>和<math>f(x)</math>中的每个解<math>y\in f(x)<math>，<math>m(x,y) , M(x,y)</math>被一个大小为<math>x</math>的多项式函数所限制，则NPO问题称为多项式有界（PB）。NPOPB 类是一类多项式有界的 NPO 问题。

如果对于每个实例<math>x</math>和<math>f(x)</math>中的每个解<math>y</math>，其测度<math>m(x,y) , M(x,y)</math>被一个大小为<math>x</math>的多项式函数所限制，则该NPO问题称为多项式有界（PB）。NPOPB 类是一类多项式有界的 NPO 问题。