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Moved page from wikipedia:en:Combinatorial optimization (history)
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[[File:Minimum spanning tree.svg|thumb|300px|right|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。找到一个最小生成树是一个涉及组合优化的常见问题。]
'''Combinatorial optimization''' is a subfield of [[mathematical optimization]] that is related to [[operations research]], [[algorithm|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.
组合优化是最优化的一个子领域，与运筹学、算法理论和计算复杂性理论有关。它在人工智能、机器学习、拍卖理论、软件工程、应用数学和理论计算机科学等领域有着重要的应用。
'''Combinatorial optimization''' <!-- synonymous or subfield?: '''discrete optimization'''{{Citation needed|date=May 2012}}--> is a topic that consists of finding an optimal object from a [[finite set]] of objects.<ref>{{harvnb|Schrijver|2003|p=1}}.</ref> In many such problems, [[exhaustive search]] is not tractable. It operates on the domain of those optimization problems in which the set of [[Candidate solution|feasible solutions]] is [[Discrete set|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|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 < ! -- 同义字段还是子字段？: 离散优化-- > 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。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。典型的问题是旅行推销员问题问题(“ TSP”)、最小生成树问题(“ MST”)和背包问题问题。
Some research literature<ref>{{cite book | title=Discrete Optimization | url=http://www.elsevier.com/locate/disopt | publisher=Elsevier | accessdate=2009-06-08}}</ref> considers [[discrete optimization]] to consist of [[integer programming]] together with combinatorial optimization (which in turn is composed of [[optimization problem]]s dealing with [[Graph (discrete mathematics)|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.
一些研究文献认为离散优化是由整数规划和组合优化组成的，尽管所有这些主题的研究文献都紧密地交织在一起。它通常涉及决定如何有效地分配用于寻找数学问题解决方案的资源。
==Applications==
Applications for combinatorial optimization include, but are not limited to:
Applications for combinatorial optimization include, but are not limited to:
组合优化的申请包括但不限于:
* [[Logistics]]<ref>{{cite journal |doi=10.1007/s10288-007-0047-3|title=Combinatorial optimization and Green Logistics|journal=4Or|volume=5|issue=2|pages=99–116|year=2007|last1=Sbihi|first1=Abdelkader|last2=Eglese|first2=Richard W.|url=https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf}}</ref>
* [[Supply chain optimization]]<ref>{{cite journal |doi=10.1016/j.omega.2015.01.006|title=Sustainable supply chain network design: An optimization-oriented review|journal=Omega|volume=54|pages=11–32|year=2015|last1=Eskandarpour|first1=Majid|last2=Dejax|first2=Pierre|last3=Miemczyk|first3=Joe|last4=Péton|first4=Olivier|url=https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%20review%20R2.pdf}}</ref>
* Developing the best airline network of spokes and destinations
* Deciding which taxis in a fleet to route to pick up fares
* Determining the optimal way to deliver packages
* Working out the best allocation of jobs to people
==Methods==
There is a large amount of literature on [[polynomial-time algorithm]]s 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 path]]s and [[shortest-path tree]]s, [[flow network|flows and circulations]], [[spanning tree]]s, [[Matching (graph theory)|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.
关于某些特殊类型的离散优化的多项式时间算法有大量的文献，相当多的文献被线性规划理论所统一。属于这个框架的组合优化问题的一些例子包括最短路径和最短路径树、流和循环、生成树、匹配和拟阵问题。
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]])
* algorithms that perform well on "random" instances (e.g. for [[Traveling salesman problem#TSP path length for random pointset in a square|TSP]])
* [[approximation algorithm]]s that run in polynomial time and find a solution that is "close" to optimal
* 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>).
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|branch-and-bound]] (an exact algorithm which can be stopped at any point in time to serve as heuristic), [[Branch and cut|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{{cn|reason=TSP is NP-hard, not NP-complete|date=March 2019}}, 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组离散项目中寻找最佳元素，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(一种可以在任何时间点停止作为启发式算法的精确算法)、分支定界法(使用线性最优化生成边界)、动态规划法(一种有限搜索窗口的递归解构法)和禁忌搜索法(一种贪婪型交换算法)。然而，遗传搜索算法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 ==
Formally, a [[combinatorial optimization]] problem <math>A</math> is a quadruple{{Citation needed|date=January 2018}} <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 a </math > 是一个四重的 < math > (i，f，m，g) </math >
* <math>I</math> is a [[Set (mathematics)|set]] of instances;
* given an instance <math>x \in I</math>, <math>f(x)</math> is the finite set of feasible solutions;
* 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]].
* <math>g</math> is the goal function, and is either <math>\min</math> or <math>\max</math>.
The goal is then to find for some instance <math>x</math> an ''optimal solution'', that is, a feasible solution <math>y</math> with
The goal is then to find for some instance <math>x</math> an optimal solution, that is, a feasible solution <math>y</math> with
然后，我们的目标是找到一个最优解，也就是一个可行的解
: <math>
<math>
《数学》
m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .
m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .
M (x，y) = g { m (x，y’) mid y’ in f (x)}。
</math>
</math>
数学
For each combinatorial optimization problem, there is a corresponding [[decision problem]] that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a [[Graph (discrete mathematics)|graph]] <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.
For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a graph <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.
对于每一个组合优化问题，都有一个相应的决策问题，它询问是否存在某一特定测度的可行解。例如，如果一个图形 < math > g </math > 包含顶点 < math > u </math > 和 < math > v </math > ，那么一个最佳化问题可能是“ find a path from < math > u </math > to < math > v </math > that uses the fewest edges”。这个问题的答案可能是，比方说，4。一个相应的决策问题是“是否存在一条从 < math > u </math > 到 < math > v </math > 使用10个或更少边的路径? ”这个问题可以用简单的“是”或“否”来回答。
In the field of [[Approximation algorithm|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.<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>
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.
在近似算法领域，算法被设计用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，因为它只具体说明了可接受的解决办法。尽管我们可以引入合适的决策问题，但这个问题更自然地被描述为一个最佳化问题问题。
== NP optimization problem ==
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.
一个 np 优化问题(NPO)是一个带有以下附加条件的组合优化优化问题。请注意，下面提到的多项式是各个函数的输入大小的函数，而不是某些隐式输入实例集的大小。
* 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 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
* <math>m</math> is [[Polynomial time|polynomial-time computable]].
This implies that the corresponding decision problem is in [[NP (complexity)|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-completeness|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 reduction|Turing]] and [[Karp reduction|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.<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>
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是 NP 完全问题。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。如果存在一个在多项式时间内找到最优解的算法，那么这个问题又称为 p 优化问题。通常，在处理类 NPO 时，人们对决策版本为 NP-complete 的优化问题感兴趣。请注意，硬度关系总是与某种程度的还原有关。由于近似算法和计算优化问题之间的联系，在某些方面保留近似值的约化比通常的图灵约化和卡普约化更受青睐。这种削减的一个例子是 l- 削减。由于这个原因，NP-complete 决策版本的优化问题不一定称为 npo 完成。
NPO is divided into the following subclasses according to their approximability:<ref name="Hromkovic02" />
NPO is divided into the following subclasses according to their approximability:
非营利组织根据其近似性可分为以下子类:
* ''NPO(I)'': Equals [[FPTAS]]. Contains the [[Knapsack problem]].
* ''NPO(II)'': Equals [[Polynomial-time approximation scheme|PTAS]]. Contains the [[Makespan]] scheduling problem.
* ''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(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(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]].
An NPO problem is called ''polynomially bounded'' (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure <math>
An NPO problem is called polynomially bounded (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure <math>
一个 NPO 问题被称为多项式有界(PB) ，如果对于每个实例 < math > x </math > 和 f (x) </math > 中的每个解 < math > y，度量值 < math >
m(x, y)
m(x, y)
M (x，y)
</math>is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.
</math>is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.
被一个 < math > x </math > 大小的多项式函数所限制。NPOPB 类是一类多项式有界的 NPO 问题。
<br />
<br />
< br/>
==Specific problems==
[[Image:TSP Deutschland 3.png|thumb|200px|An optimal traveling salesperson tour through [[Germany]]’s 15 largest cities. It is the shortest among 43,589,145,600<ref>Take one city, and take all possible orders of the other 14 cities. Then divide by two because it does not matter in which direction in time they come after each other: 14!/2 = 43,589,145,600.</ref> possible tours visiting each city exactly once.]]
An optimal traveling salesperson tour through [[Germany’s 15 largest cities. It is the shortest among 43,589,145,600 possible tours visiting each city exactly once.]]
最佳的旅行推销员之旅[德国最大的15个城市。在43,589,145,600个可能的游览每个城市的旅游团中，它是最短的
* [[Assignment problem]]
* [[Closure problem]]
* [[Constraint satisfaction problem]]
* [[Cutting stock problem]]
*[[Dominating set]] problem
* [[Integer programming]]
* [[Knapsack problem]]
*[[Minimum relevant variables in linear system]]
*[[Minimum spanning tree]]
* [[Nurse scheduling problem]]
*[[Set cover problem]]
* [[Traveling salesman problem]]
* [[Vehicle rescheduling problem]]
* [[Vehicle routing problem]]
* [[Weapon target assignment problem]]
==See also==
*[[Constraint composite graph]]
==Notes==
{{reflist}}
==References==
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Http://www.tsp.gatech.edu/optimal/index.html
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| publisher = [[University of Waterloo]]
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2012年3月24日 | publisher = 滑铁卢大学
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2016年
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}} ''(Information on the largest TSP instances solved to date.)''
}} (Information on the largest TSP instances solved to date.)
}(迄今为止已解决的最大 TSP 实例的信息)
*{{Cite web
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1 = Crescenzi
| editor-first1 = Pierluigi
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2 = Kann
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| editor-last5 = Woeginger
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| editor-first5 = Gerhard
| editor-first5 = Gerhard
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| title = A Compendium of NP Optimization Problems
| title = A Compendium of NP Optimization Problems
最优化问题概要
| ref = harv
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}} ''(This is a continuously updated catalog of approximability results for NP optimization problems.)''
}} (This is a continuously updated catalog of approximability results for NP optimization problems.)
}(这是一个不断更新的目录近似结果的 NP 优化问题。)
*{{Cite book
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| editor-first1 = Arnab
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1 = Arnab
| editor-last2 = Chakrabarti
| editor-last2 = Chakrabarti
2 = Chakrabarti
| editor-first2 = Bikas K
| editor-first2 = Bikas K
2 = Bikas k
| editor2-link = Bikas K Chakrabarti
| editor2-link = Bikas K Chakrabarti
2-link = Bikas k Chakrabarti
| title = Quantum Annealing and Related Optimization Methods
| title = Quantum Annealing and Related Optimization Methods
量子退火和相关的优化方法
| series = Lecture Notes in Physics
| series = Lecture Notes in Physics
| 系列 = 物理学讲义
| volume = 679
| volume = 679
679
| publisher = Springer
| publisher = Springer
| publisher = Springer
| year = 2005
| year = 2005
2005年
| ref = harv
| ref = harv
= harv
| bibcode = 2005qnro.book.....D
| bibcode = 2005qnro.book.....D
2005 qnro. book... d
}}
}}
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*{{Cite journal
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| last1 = Das
1 = Das
| first1 = Arnab
| first1 = Arnab
1 = Arnab
| last2 = Chakrabarti
| last2 = Chakrabarti
2 = Chakrabarti
| first2 = Bikas K
| first2 = Bikas K
2 = Bikas k
| title = Colloquium: Quantum annealing and analog quantum computation
| title = Colloquium: Quantum annealing and analog quantum computation
| title = 学术讨论会: 量子退火和模拟量子计算
| journal = Rev. Mod. Phys.
| journal = Rev. Mod. Phys.
| 日记 = rev。Mod.女名女子名。
| volume = 80
| volume = 80
80
| issue = 3
| issue = 3
第三期
| page = 1061
| page = 1061
1061
| year = 2008
| year = 2008
2008年
| doi = 10.1103/RevModPhys.80.1061
| doi = 10.1103/RevModPhys.80.1061
| doi = 10.1103/RevModPhys. 80.1061
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= harv
| citeseerx = 10.1.1.563.9990
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10.1.1.563.9990
| bibcode = 2008RvMP...80.1061D
| bibcode = 2008RvMP...80.1061D
2008/rvmp... 80.1061 d
| arxiv = 0801.2193
| arxiv = 0801.2193
0801.2193
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}}
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*{{Cite book
| last = Lawler
| last = Lawler
| last = Lawler
| first = Eugene
| first = Eugene
第一 = 尤金
| author-link = Eugene Lawler
| author-link = Eugene Lawler
| 作者链接 = Eugene Lawler
| title = Combinatorial Optimization: Networks and Matroids
| title = Combinatorial Optimization: Networks and Matroids
组合优化: 网络与拟阵
| year = 2001
| year = 2001
2001年
| publisher = Dover
| publisher = Dover
| publisher = Dover
| isbn = 0-486-41453-1
| isbn = 0-486-41453-1
| isbn = 0-486-41453-1
| <!-- pages = 117–120 -->
| <!-- pages = 117–120 -->
| < ! -- pages = 117-120 -- >
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}}
}}
}}
*{{Cite book
| first = Jon
| first = Jon
| first = Jon
| last = Lee
| last = Lee
| last = Lee
| author-link = Jon Lee (mathematician)
| author-link = Jon Lee (mathematician)
乔恩 · 李(数学家)
| url = https://books.google.com/books?id=3pL1B7WVYnAC
| url = https://books.google.com/books?id=3pL1B7WVYnAC
Https://books.google.com/books?id=3pl1b7wvynac
| title = A First Course in Combinatorial Optimization
| title = A First Course in Combinatorial Optimization
组合优化的第一堂课
| publisher = Cambridge University Press
| publisher = Cambridge University Press
剑桥大学出版社
| year = 2004
| year = 2004
2004年
| isbn = 0-521-01012-8
| isbn = 0-521-01012-8
| isbn = 0-521-01012-8
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}}
}}
}}
*{{Cite book
| last1 = Papadimitriou
| last1 = Papadimitriou
1 = Papadimitriou
| first1 = Christos H.
| first1 = Christos H.
1 = Christos h.
| last2 = Steiglitz
| last2 = Steiglitz
2 = Steiglitz
| first2 = Kenneth
| first2 = Kenneth
2 = Kenneth
| author2-link = Kenneth Steiglitz
| author2-link = Kenneth Steiglitz
| author2-link = Kenneth Steiglitz
| title = Combinatorial Optimization : Algorithms and Complexity
| title = Combinatorial Optimization : Algorithms and Complexity
组合优化: 算法与复杂性
| publisher = Dover
| publisher = Dover
| publisher = Dover
| date = July 1998
| date = July 1998
日期 = 1998年7月
| isbn = 0-486-40258-4
| isbn = 0-486-40258-4
| isbn = 0-486-40258-4
| ref = harv
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}}
}}
}}
*{{Cite book
| last = Schrijver
| last = Schrijver
| last = Schrijver
| first = Alexander
| first = Alexander
第一个 = 亚历山大
| title = Combinatorial Optimization: Polyhedra and Efficiency
| title = Combinatorial Optimization: Polyhedra and Efficiency
组合优化: 多面体与效率
| publisher = Springer
| publisher = Springer
| publisher = Springer
| series = Algorithms and Combinatorics
| series = Algorithms and Combinatorics
序列 = 算法和组合数学
| volume = 24
| volume = 24
24
| year = 2003
| year = 2003
2003年
| ref = harv
| ref = harv
= harv
| url = https://books.google.com/books?id=mqGeSQ6dJycC
| url = https://books.google.com/books?id=mqGeSQ6dJycC
Https://books.google.com/books?id=mqgesq6djycc
| isbn = 9783540443896
| isbn = 9783540443896
9783540443896
}}
}}
}}
*{{Cite book
| last = Schrijver
| last = Schrijver
| last = Schrijver
| first = Alexander
| first = Alexander
第一个 = 亚历山大
| chapter = On the history of combinatorial optimization (till 1960)
| chapter = On the history of combinatorial optimization (till 1960)
| 第二章: 组合优化的历史(1960年以前)
| title = Handbook of Discrete Optimization
| title = Handbook of Discrete Optimization
| title = 离散优化手册
| editor-last1 = Aardal
| editor-last1 = Aardal
1 = Aardal
| editor-first1 = K.|editor1-link=Karen Aardal
| editor-first1 = K.|editor1-link=Karen Aardal
1 = k | editor1-link = Karen Aardal
| editor-last2 = Nemhauser
| editor-last2 = Nemhauser
2 = Nemhauser
| editor-first2 = G.L.
| editor-first2 = G.L.
2 = g.l.
| editor-last3 = Weismantel
| editor-last3 = Weismantel
3 = Weismantel
| editor-first3 = R.
| editor-first3 = R.
3 = r.
| publisher = Elsevier
| publisher = Elsevier
| publisher = Elsevier
| year = 2005
| year = 2005
2005年
| pages = 1–68
| pages = 1–68
| 页数 = 1-68
| chapter-url = http://homepages.cwi.nl/~lex/files/histco.pdf
| chapter-url = http://homepages.cwi.nl/~lex/files/histco.pdf
| chapter-url = http://homepages.cwi.nl/~lex/files/histco.pdf
| ref = harv
| ref = harv
= harv
}}
}}
}}
*{{Cite book
| last = Schrijver
| last = Schrijver
| last = Schrijver
| first = Alexander
| first = Alexander
第一个 = 亚历山大
| title = A Course in Combinatorial Optimization
| title = A Course in Combinatorial Optimization
文章标题: 组合优化课程
| url = http://homepages.cwi.nl/~lex/files/dict.pdf
| url = http://homepages.cwi.nl/~lex/files/dict.pdf
Http://homepages.cwi.nl/~lex/files/dict.pdf
| date = February 1, 2006
| date = February 1, 2006
日期 = 2006年2月1日
| ref = harv
| ref = harv
= harv
}}
}}
}}
*{{Cite book
| last1 = Sierksma
| last1 = Sierksma
1 = Sierksma
| first1 = Gerard
| first1 = Gerard
1 = Gerard
| last2 = Ghosh
| last2 = Ghosh
2 = Ghosh
| first2 = Diptesh
| first2 = Diptesh
2 = Diptesh
| author1-link = Gerard Sierksma
| author1-link = Gerard Sierksma
1-link = Gerard Sierksma
| title = Networks in Action; Text and Computer Exercises in Network Optimization
| title = Networks in Action; Text and Computer Exercises in Network Optimization
行动中的网络; 网络优化中的文本和计算机练习
| publisher = Springer
| publisher = Springer
| publisher = Springer
| date = 2010
| date = 2010
2010年
| isbn = 978-1-4419-5512-8
| isbn = 978-1-4419-5512-8
| isbn = 978-1-4419-5512-8
| ref = harv
| ref = harv
= harv
}}
}}
}}
*{{Cite book
| author1=Gerard Sierksma
| author1=Gerard Sierksma
1 = Gerard Sierksma
| author2=Yori Zwols
| author2=Yori Zwols
2 = Yori Zwols
| title=Linear and Integer Optimization: Theory and Practice
| title=Linear and Integer Optimization: Theory and Practice
| title = 线性和整数优化: 理论与实践
| year=2015
| year=2015
2015年
| publisher=CRC Press
| publisher=CRC Press
| publisher = CRC Press
| isbn=978-1-498-71016-9
| isbn=978-1-498-71016-9
| isbn = 978-1-498-71016-9
}}
}}
}}
*{{Cite book
| last = Pintea
| last = Pintea
| last = pinter
| first = C-M.
| first = C-M.
| first = C-M.
| title = Advances in Bio-inspired Computing for Combinatorial Optimization Problem
| title = Advances in Bio-inspired Computing for Combinatorial Optimization Problem
| title = 组合优化问题的仿生计算进展
| url = https://www.springer.com/la/book/9783642401787
| url = https://www.springer.com/la/book/9783642401787
Https://www.springer.com/la/book/9783642401787
| publisher = Springer
| publisher = Springer
| publisher = Springer
| year = 2014
| year = 2014
2014年
| isbn = 978-3-642-40178-7
| isbn = 978-3-642-40178-7
| isbn = 978-3-642-40178-7
| ref = harv
| ref = harv
= harv
| series = Intelligent Systems Reference Library
| series = Intelligent Systems Reference Library
| series = 智能系统参考图书馆
}}
}}
}}
==External links==
{{Commonscat}}
*[https://www.springer.com/mathematics/journal/10878 Journal of Combinatorial Optimization]
*[http://www.iasi.cnr.it/aussois The Aussois Combinatorial Optimization Workshop]
*[http://sourceforge.net/projects/jcop/ Java Combinatorial Optimization Platform] (open source code)
*[https://www.mjc2.com/staff-planning-complexity.htm Why is scheduling people hard?]
*[https://www7.in.tum.de/~kugele/files/jobsis.pdf Complexity classes for optimization problems / Stefan Kugele]
{{DEFAULTSORT:Combinatorial Optimization}}
[[Category:Combinatorial optimization| ]]
[[Category:Computational complexity theory]]
Category:Computational complexity theory
类别: 计算复杂性理论
[[Category:Theoretical computer science]]
Category:Theoretical computer science
类别: 理论计算机科学
[[eo:Diskreta optimumigo]]
eo:Diskreta optimumigo
2: Diskreta optiumigo
<noinclude>
<small>This page was moved from [[wikipedia:en:Combinatorial optimization]]. Its edit history can be viewed at [[组合优化/edithistory]]</small></noinclude>
[[Category:待整理页面]]
[[File:Minimum spanning tree.svg|thumb|300px|right|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。找到一个最小生成树是一个涉及组合优化的常见问题。]
'''Combinatorial optimization''' is a subfield of [[mathematical optimization]] that is related to [[operations research]], [[algorithm|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.
组合优化是最优化的一个子领域，与运筹学、算法理论和计算复杂性理论有关。它在人工智能、机器学习、拍卖理论、软件工程、应用数学和理论计算机科学等领域有着重要的应用。
'''Combinatorial optimization''' <!-- synonymous or subfield?: '''discrete optimization'''{{Citation needed|date=May 2012}}--> is a topic that consists of finding an optimal object from a [[finite set]] of objects.<ref>{{harvnb|Schrijver|2003|p=1}}.</ref> In many such problems, [[exhaustive search]] is not tractable. It operates on the domain of those optimization problems in which the set of [[Candidate solution|feasible solutions]] is [[Discrete set|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|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 < ! -- 同义字段还是子字段？: 离散优化-- > 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。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。典型的问题是旅行推销员问题问题(“ TSP”)、最小生成树问题(“ MST”)和背包问题问题。
Some research literature<ref>{{cite book | title=Discrete Optimization | url=http://www.elsevier.com/locate/disopt | publisher=Elsevier | accessdate=2009-06-08}}</ref> considers [[discrete optimization]] to consist of [[integer programming]] together with combinatorial optimization (which in turn is composed of [[optimization problem]]s dealing with [[Graph (discrete mathematics)|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.
一些研究文献认为离散优化是由整数规划和组合优化组成的，尽管所有这些主题的研究文献都紧密地交织在一起。它通常涉及决定如何有效地分配用于寻找数学问题解决方案的资源。
==Applications==
Applications for combinatorial optimization include, but are not limited to:
Applications for combinatorial optimization include, but are not limited to:
组合优化的申请包括但不限于:
* [[Logistics]]<ref>{{cite journal |doi=10.1007/s10288-007-0047-3|title=Combinatorial optimization and Green Logistics|journal=4Or|volume=5|issue=2|pages=99–116|year=2007|last1=Sbihi|first1=Abdelkader|last2=Eglese|first2=Richard W.|url=https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf}}</ref>
* [[Supply chain optimization]]<ref>{{cite journal |doi=10.1016/j.omega.2015.01.006|title=Sustainable supply chain network design: An optimization-oriented review|journal=Omega|volume=54|pages=11–32|year=2015|last1=Eskandarpour|first1=Majid|last2=Dejax|first2=Pierre|last3=Miemczyk|first3=Joe|last4=Péton|first4=Olivier|url=https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%20review%20R2.pdf}}</ref>
* Developing the best airline network of spokes and destinations
* Deciding which taxis in a fleet to route to pick up fares
* Determining the optimal way to deliver packages
* Working out the best allocation of jobs to people
==Methods==
There is a large amount of literature on [[polynomial-time algorithm]]s 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 path]]s and [[shortest-path tree]]s, [[flow network|flows and circulations]], [[spanning tree]]s, [[Matching (graph theory)|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.
关于某些特殊类型的离散优化的多项式时间算法有大量的文献，相当多的文献被线性规划理论所统一。属于这个框架的组合优化问题的一些例子包括最短路径和最短路径树、流和循环、生成树、匹配和拟阵问题。
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]])
* algorithms that perform well on "random" instances (e.g. for [[Traveling salesman problem#TSP path length for random pointset in a square|TSP]])
* [[approximation algorithm]]s that run in polynomial time and find a solution that is "close" to optimal
* 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>).
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|branch-and-bound]] (an exact algorithm which can be stopped at any point in time to serve as heuristic), [[Branch and cut|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{{cn|reason=TSP is NP-hard, not NP-complete|date=March 2019}}, 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组离散项目中寻找最佳元素，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(一种可以在任何时间点停止作为启发式算法的精确算法)、分支定界法(使用线性最优化生成边界)、动态规划法(一种有限搜索窗口的递归解构法)和禁忌搜索法(一种贪婪型交换算法)。然而，遗传搜索算法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 ==
Formally, a [[combinatorial optimization]] problem <math>A</math> is a quadruple{{Citation needed|date=January 2018}} <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 a </math > 是一个四重的 < math > (i，f，m，g) </math >
* <math>I</math> is a [[Set (mathematics)|set]] of instances;
* given an instance <math>x \in I</math>, <math>f(x)</math> is the finite set of feasible solutions;
* 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]].
* <math>g</math> is the goal function, and is either <math>\min</math> or <math>\max</math>.
The goal is then to find for some instance <math>x</math> an ''optimal solution'', that is, a feasible solution <math>y</math> with
The goal is then to find for some instance <math>x</math> an optimal solution, that is, a feasible solution <math>y</math> with
然后，我们的目标是找到一个最优解，也就是一个可行的解
: <math>
<math>
《数学》
m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .
m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .
M (x，y) = g { m (x，y’) mid y’ in f (x)}。
</math>
</math>
数学
For each combinatorial optimization problem, there is a corresponding [[decision problem]] that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a [[Graph (discrete mathematics)|graph]] <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.
For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a graph <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.
对于每一个组合优化问题，都有一个相应的决策问题，它询问是否存在某一特定测度的可行解。例如，如果一个图形 < math > g </math > 包含顶点 < math > u </math > 和 < math > v </math > ，那么一个最佳化问题可能是“ find a path from < math > u </math > to < math > v </math > that uses the fewest edges”。这个问题的答案可能是，比方说，4。一个相应的决策问题是“是否存在一条从 < math > u </math > 到 < math > v </math > 使用10个或更少边的路径? ”这个问题可以用简单的“是”或“否”来回答。
In the field of [[Approximation algorithm|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.<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>
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.
在近似算法领域，算法被设计用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，因为它只具体说明了可接受的解决办法。尽管我们可以引入合适的决策问题，但这个问题更自然地被描述为一个最佳化问题问题。
== NP optimization problem ==
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.
一个 np 优化问题(NPO)是一个带有以下附加条件的组合优化优化问题。请注意，下面提到的多项式是各个函数的输入大小的函数，而不是某些隐式输入实例集的大小。
* 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 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
* <math>m</math> is [[Polynomial time|polynomial-time computable]].
This implies that the corresponding decision problem is in [[NP (complexity)|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-completeness|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 reduction|Turing]] and [[Karp reduction|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.<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>
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是 NP 完全问题。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。如果存在一个在多项式时间内找到最优解的算法，那么这个问题又称为 p 优化问题。通常，在处理类 NPO 时，人们对决策版本为 NP-complete 的优化问题感兴趣。请注意，硬度关系总是与某种程度的还原有关。由于近似算法和计算优化问题之间的联系，在某些方面保留近似值的约化比通常的图灵约化和卡普约化更受青睐。这种削减的一个例子是 l- 削减。由于这个原因，NP-complete 决策版本的优化问题不一定称为 npo 完成。
NPO is divided into the following subclasses according to their approximability:<ref name="Hromkovic02" />
NPO is divided into the following subclasses according to their approximability:
非营利组织根据其近似性可分为以下子类:
* ''NPO(I)'': Equals [[FPTAS]]. Contains the [[Knapsack problem]].
* ''NPO(II)'': Equals [[Polynomial-time approximation scheme|PTAS]]. Contains the [[Makespan]] scheduling problem.
* ''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(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(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]].
An NPO problem is called ''polynomially bounded'' (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure <math>
An NPO problem is called polynomially bounded (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure <math>
一个 NPO 问题被称为多项式有界(PB) ，如果对于每个实例 < math > x </math > 和 f (x) </math > 中的每个解 < math > y，度量值 < math >
m(x, y)
m(x, y)
M (x，y)
</math>is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.
</math>is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.
被一个 < math > x </math > 大小的多项式函数所限制。NPOPB 类是一类多项式有界的 NPO 问题。
<br />
<br />
< br/>
==Specific problems==
[[Image:TSP Deutschland 3.png|thumb|200px|An optimal traveling salesperson tour through [[Germany]]’s 15 largest cities. It is the shortest among 43,589,145,600<ref>Take one city, and take all possible orders of the other 14 cities. Then divide by two because it does not matter in which direction in time they come after each other: 14!/2 = 43,589,145,600.</ref> possible tours visiting each city exactly once.]]
An optimal traveling salesperson tour through [[Germany’s 15 largest cities. It is the shortest among 43,589,145,600 possible tours visiting each city exactly once.]]
最佳的旅行推销员之旅[德国最大的15个城市。在43,589,145,600个可能的游览每个城市的旅游团中，它是最短的
* [[Assignment problem]]
* [[Closure problem]]
* [[Constraint satisfaction problem]]
* [[Cutting stock problem]]
*[[Dominating set]] problem
* [[Integer programming]]
* [[Knapsack problem]]
*[[Minimum relevant variables in linear system]]
*[[Minimum spanning tree]]
* [[Nurse scheduling problem]]
*[[Set cover problem]]
* [[Traveling salesman problem]]
* [[Vehicle rescheduling problem]]
* [[Vehicle routing problem]]
* [[Weapon target assignment problem]]
==See also==
*[[Constraint composite graph]]
==Notes==
{{reflist}}
==References==
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| title = Handbook of Discrete Optimization
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| title = 离散优化手册
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==External links==
{{Commonscat}}
*[https://www.springer.com/mathematics/journal/10878 Journal of Combinatorial Optimization]
*[http://www.iasi.cnr.it/aussois The Aussois Combinatorial Optimization Workshop]
*[http://sourceforge.net/projects/jcop/ Java Combinatorial Optimization Platform] (open source code)
*[https://www.mjc2.com/staff-planning-complexity.htm Why is scheduling people hard?]
*[https://www7.in.tum.de/~kugele/files/jobsis.pdf Complexity classes for optimization problems / Stefan Kugele]
{{DEFAULTSORT:Combinatorial Optimization}}
[[Category:Combinatorial optimization| ]]
[[Category:Computational complexity theory]]
Category:Computational complexity theory
类别: 计算复杂性理论
[[Category:Theoretical computer science]]
Category:Theoretical computer science
类别: 理论计算机科学
[[eo:Diskreta optimumigo]]
eo:Diskreta optimumigo
2: Diskreta optiumigo
<noinclude>
<small>This page was moved from [[wikipedia:en:Combinatorial optimization]]. Its edit history can be viewed at [[组合优化/edithistory]]</small></noinclude>
[[Category:待整理页面]]