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第237行： − ''''''指派问题 Assignment Problem ''''''+ − ''''''封闭性问题 Closure Problem ''''''+ − ''''''约束满足问题 Constraint Satisfaction Problem ''''''+ − ''''''切割问题 Cutting Stock Problem ''''''+ − ''''''控制集问题 Dominating Set '''''' + − ''''''整数规划 Integer Programming ''''''+ − ''''''背包问题 Knapsack Problem ''''''+ − ''''''线性系统中的最小相关变量 Minimum Relevant Variables In Linear System ''''''+ − ''''''最小生成树 Minimum Spanning Tree ''''''+ − ''''''护士调度问题 Nurse Scheduling Problem ''''''+ − ''''''集合覆盖问题 Set Cover Problem ''''''+ − ''''''旅行商问题 Traveling Salesman Problem ''''''+ − ''''''车辆重新调度问题 Vehicle Rescheduling Problem ''''''+ − ''''''车辆线路优化问题 Vehicle Routing Problem ''''''+ − ''''''武器目标分配问题 Weapon Target Assignment Problem ''''''+
无编辑摘要
A [[minimum spanning tree of a weighted planar graph. Finding a minimum spanning tree is a common problem involving combinatorial optimization.]]
A [[minimum spanning tree of a weighted planar graph. Finding a minimum spanning tree is a common problem involving combinatorial optimization.]]
一个加权平面图的''''''最小生成树 Minimum Spanning Tree ''''''。找到最小生成树是一个涉及''''''组合优化 Combinatorial Optimization ''''''的常见问题。
一个加权平面图的'''最小生成树 Minimum Spanning Tree '''。找到最小生成树是一个涉及'''组合优化 Combinatorial Optimization '''的常见问题。
Combinatorial optimization is a subfield of mathematical optimization that is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, applied mathematics and theoretical computer science.
Combinatorial optimization is a subfield of mathematical optimization that is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, applied mathematics and theoretical computer science.
组合优化是''''''数学优化方法 Mathematical Optimization ''''''的一个子领域，与''''''运筹学 Operations Research ''''''、''''''算法理论 Algorithm Theory ''''''和''''''计算复杂性理论 Computational Complexity ''''''有关。它在''''''人工智能 Artificial Intelligence ''''''、''''''机器学习 Machine Learning ''''''、''''''拍卖理论 Auction Theory ''''''、''''''软件工程 Software Engineering ''''''、''''''应用数学 Applied Mathematics ''''''和''''''理论计算机科学 Theoretical Computer Science ''''''等领域有着重要的应用。
组合优化是'''数学优化方法 Mathematical Optimization '''的一个子领域，与'''运筹学 Operations Research '''、'''算法理论 Algorithm Theory '''和'''计算复杂性理论 Computational Complexity '''有关。它在'''人工智能 Artificial Intelligence '''、'''机器学习 Machine Learning '''、'''拍卖理论 Auction Theory '''、'''软件工程 Software Engineering '''、'''应用数学 Applied Mathematics '''和'''理论计算机科学 Theoretical Computer Science '''等领域有着重要的应用。
'''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，''''''穷举搜索 exhaustive search ''''''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''''''[[旅行商问题]] Traveling Salesman Problem ''''''(“ TSP”)、[[最小生成树问题]](“ MST”)和''''''[[背包问题]] Knapsack Problem ''''''。
'''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，'''穷举搜索 exhaustive search '''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'''[[旅行商问题]] Traveling Salesman Problem '''(“ TSP”)、[[最小生成树问题]](“ MST”)和'''[[背包问题]] Knapsack Problem '''。
Some research literature considers discrete optimization to consist of integer programming together with combinatorial optimization (which in turn is composed of optimization problems dealing with graph structures) although all of these topics have closely intertwined research literature. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems.
Some research literature considers discrete optimization to consist of integer programming together with combinatorial optimization (which in turn is composed of optimization problems dealing with graph structures) although all of these topics have closely intertwined research literature. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems.
一些研究文献<ref>{{cite book | title=Discrete Optimization | url=http://www.elsevier.com/locate/disopt | publisher=Elsevier | accessdate=2009-06-08}}</ref>认为''''''离散优化 Discrete Optimization ''''''是由''''''整数规划 Integer Programming ''''''和组合优化组成的（反过来由解决图结构的优化问题组成），尽管所有这些主题的研究文献都紧密地交织在一起。它通常涉及如何有效地分配用于寻找数学问题解决方案的资源的决策。
一些研究文献<ref>{{cite book | title=Discrete Optimization | url=http://www.elsevier.com/locate/disopt | publisher=Elsevier | accessdate=2009-06-08}}</ref>认为'''离散优化 Discrete Optimization '''是由'''整数规划 Integer Programming '''和组合优化组成的（反过来由解决图结构的优化问题组成），尽管所有这些主题的研究文献都紧密地交织在一起。它通常涉及如何有效地分配用于寻找数学问题解决方案的资源的决策。
There is a large amount of literature on polynomial-time algorithms for certain special classes of discrete optimization, a considerable amount of it unified by the theory of linear programming. Some examples of combinatorial optimization problems that fall into this framework are shortest paths and shortest-path trees, flows and circulations, spanning trees, matching, and matroid problems.
There is a large amount of literature on polynomial-time algorithms for certain special classes of discrete optimization, a considerable amount of it unified by the theory of linear programming. Some examples of combinatorial optimization problems that fall into this framework are shortest paths and shortest-path trees, flows and circulations, spanning trees, matching, and matroid problems.
对于某些特殊的离散优化问题，有大量的文献是关于''''''多项式时间算法 Polynomial-Time Algorithm ''''''的，其中相当一部分是通过''''''线性规划 Linear Programming ''''''理论统一起来的。属于这个框架的组合优化问题的一些例子包括''''''最短路径 Shortest paths ''''''和''''''最短路径树 Shortest-path Tree ''''''、''''''流和循环 Flows And Circulations ''''''、生成树、''''''匹配和拟阵 Matching And Matroid Problems ''''''问题。
对于某些特殊的离散优化问题，有大量的文献是关于'''多项式时间算法 Polynomial-Time Algorithm '''的，其中相当一部分是通过'''线性规划 Linear Programming '''理论统一起来的。属于这个框架的组合优化问题的一些例子包括'''最短路径 Shortest paths '''和'''最短路径树 Shortest-path Tree '''、'''流和循环 Flows And Circulations '''、生成树、'''匹配和拟阵 Matching And Matroid Problems '''问题。
For NP-complete discrete optimization problems, current research literature includes the following topics:
For NP-complete discrete optimization problems, current research literature includes the following topics:
对于'''''' NP完全 NP-Complete ''''''的离散优化问题，目前的研究文献包括以下主题:
对于''' NP完全 NP-Complete '''的离散优化问题，目前的研究文献包括以下主题:
* 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]])
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）
* [[approximation algorithm]]s that run in polynomial time and find a solution that is "close" to optimal
* [[approximation algorithm]]s that run in polynomial time and find a solution that is "close" to optimal
'''近似算法 Approximation Algorithm '''在多项式时间内运行并找到一个“接近”最优值的解
* solving real-world instances that arise in practice and do not necessarily exhibit the worst-case behavior inherent in NP-complete problems (e.g. TSP instances with tens of thousands of nodes<ref>{{harvnb|Cook|2016}}.</ref>).
* solving real-world instances that arise in practice and do not necessarily exhibit the worst-case behavior inherent in NP-complete problems (e.g. TSP instances with tens of thousands of nodes<ref>{{harvnb|Cook|2016}}.</ref>).
解决现实世界中出现的实例，这些实例不一定表现出NP完全问题固有的最坏情况（例如，具有成千上万个节点的TSP实例<ref>{{harvnb|Cook|2016}}.</ref>)
解决现实世界中出现的实例，这些实例不一定表现出NP完全问题固有的最坏情况（例如，具有成千上万个节点的TSP实例<ref>{{harvnb|Cook|2016}}.</ref>)
Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of search algorithm or metaheuristic can be used to solve them. Perhaps the most universally applicable approaches are branch-and-bound (an exact algorithm which can be stopped at any point in time to serve as heuristic), branch-and-cut (uses linear optimisation to generate bounds), dynamic programming (a recursive solution construction with limited search window) and tabu search (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are NP-complete, such as the traveling salesman problem, this is expected unless P=NP.
Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of search algorithm or metaheuristic can be used to solve them. Perhaps the most universally applicable approaches are branch-and-bound (an exact algorithm which can be stopped at any point in time to serve as heuristic), branch-and-cut (uses linear optimisation to generate bounds), dynamic programming (a recursive solution construction with limited search window) and tabu search (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are NP-complete, such as the traveling salesman problem, this is expected unless P=NP.
组合优化问题可以看作是在一些离散项目中搜索最佳元素，因此，原则上，任何一种搜索算法或元启发式算法都可以用来解决它们。也许最普遍适用的方法是''''''分支定界法 Branch-and-bound ''''''(一种可以在任何时间点停止来用作启发式的精确算法)、''''''分支切割法 Branch-and-cut ''''''(使用线性最优化生成边界)、''''''动态规划法 Dynamic Programming ''''''(一种有限搜索窗口的递归解构法)和''''''禁忌搜索法 Tabu Search ''''''(一种贪婪交换算法)。然而，''''''遗传搜索算法 Generic Search Algorithms ''''''不能保证首先找到最优解，也不能保证快速运行(在多项式时间内)。由于一些离散优化问题是NP完全的，例如旅行商问题，除非P=NP，否则这是可以预期的。
组合优化问题可以看作是在一些离散项目中搜索最佳元素，因此，原则上，任何一种搜索算法或元启发式算法都可以用来解决它们。也许最普遍适用的方法是'''分支定界法 Branch-and-bound '''(一种可以在任何时间点停止来用作启发式的精确算法)、'''分支切割法 Branch-and-cut '''(使用线性最优化生成边界)、'''动态规划法 Dynamic Programming '''(一种有限搜索窗口的递归解构法)和'''禁忌搜索法 Tabu Search '''(一种贪婪交换算法)。然而，'''遗传搜索算法 Generic Search Algorithms '''不能保证首先找到最优解，也不能保证快速运行(在多项式时间内)。由于一些离散优化问题是NP完全的，例如旅行商问题，除非P=NP，否则这是可以预期的。
== Formal definition 形式化定义 ==
== Formal definition 形式化定义 ==
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'''''' 近似算法approximation algorithms''''''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>
在''' 近似算法approximation algorithms'''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>
== NP optimization problem NP优化问题==
== NP optimization problem NP优化问题==
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''''''P-优化（PO）问题 P-optimization problem ''''''。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''''''图灵和卡普规约 Turing and Karp Reductions ''''''更为可取。这种规约的一个例子就是''''''L-规约 L-reduction ''''''。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'''P-优化（PO）问题 P-optimization problem '''。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'''图灵和卡普规约 Turing and Karp Reductions '''更为可取。这种规约的一个例子就是'''L-规约 L-reduction '''。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''''''完全多项式时间近似方案 Fully Polynomial-time approximation scheme | PTAS ''''''。包含背包问题。
''NPO(I)''：等价于'''完全多项式时间近似方案 Fully Polynomial-time approximation scheme | PTAS '''。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 Polynomial-time approximation scheme | PTAS '''''' 。包含分批调度问题。
''NPO(II)''：等价于'''多项式时间近似方案 Polynomial-time approximation scheme | PTAS ''' 。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（approximable）。包含''''''最大可满足性问题 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'''最大可满足性问题 MAX-SAT '''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)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio bounded by some function on n. In Hromkovic's book, all NPO(IV)-problems are excluded from this class unless P=NP. Contains the [[Travelling salesman problem|TSP]] and [[Clique problem|Max Clique problems]].
* ''NPO(V)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio bounded by some function on n. In Hromkovic's book, all NPO(IV)-problems are excluded from this class unless P=NP. Contains the [[Travelling salesman problem|TSP]] and [[Clique problem|Max Clique problems]].
''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''''''团问题|最大团问题 Clique problem|Max Clique problems ''''''。
''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'''团问题|最大团问题 Clique problem|Max Clique problems '''。
* [[Assignment problem]]
* [[Assignment problem]]
'''指派问题 Assignment Problem '''
* [[Closure problem]]
* [[Closure problem]]
'''封闭性问题 Closure Problem '''
* [[Constraint satisfaction problem]]
* [[Constraint satisfaction problem]]
'''约束满足问题 Constraint Satisfaction Problem '''
* [[Cutting stock problem]]
* [[Cutting stock problem]]
'''切割问题 Cutting Stock Problem '''
*[[Dominating set]] problem
*[[Dominating set]] problem
'''控制集问题 Dominating Set '''
* [[Integer programming]]
* [[Integer programming]]
'''整数规划 Integer Programming '''
* [[Knapsack problem]]
* [[Knapsack problem]]
'''背包问题 Knapsack Problem '''
*[[Minimum relevant variables in linear system]]
*[[Minimum relevant variables in linear system]]
'''线性系统中的最小相关变量 Minimum Relevant Variables In Linear System '''
*[[Minimum spanning tree]]
*[[Minimum spanning tree]]
'''最小生成树 Minimum Spanning Tree '''
* [[Nurse scheduling problem]]
* [[Nurse scheduling problem]]
'''护士调度问题 Nurse Scheduling Problem '''
*[[Set cover problem]]
*[[Set cover problem]]
'''集合覆盖问题 Set Cover Problem '''
* [[Traveling salesman problem]]
* [[Traveling salesman problem]]
'''旅行商问题 Traveling Salesman Problem '''
* [[Vehicle rescheduling problem]]
* [[Vehicle rescheduling problem]]
'''车辆重新调度问题 Vehicle Rescheduling Problem '''
* [[Vehicle routing problem]]
* [[Vehicle routing problem]]
'''车辆线路优化问题 Vehicle Routing Problem '''
* [[Weapon target assignment problem]]
* [[Weapon target assignment problem]]
'''武器目标分配问题 Weapon Target Assignment Problem '''
==See also==
==See also==