组合优化

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文件:Minimum spanning tree.svg
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。


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 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 理论计算机科学 Theoretical Computer Science 等领域有着重要的应用。


Combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects.[1] 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 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.

组合优化主要是从一个有限的对象集合中寻找一个最佳对象。[2]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

-- Flipped讨论) 第一句话读不通顺


Some research literature[3] 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[4]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。


Applications 应用

Applications for combinatorial optimization include, but are not limited to:

Applications for combinatorial optimization include, but are not limited to:

组合优化的应用包括但不限于:


物流[6]

供应链优化[8]

  • Developing the best airline network of spokes and destinations

发展最好辐条和目的地的航空公司网络

  • Deciding which taxis in a fleet to route to pick up fares

决定车队中哪辆出租车来载客

  • Determining the optimal way to deliver packages

确定运送包裹的最佳方式

  • Working out the best allocation of jobs to people

制定最佳工作分配

Methods 方法

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 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:

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 TSP)

在“随机”实例上表现良好的算法(例如, TSP)

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[9]).

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[10])


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模板:Cn, 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。

Formal definition 形式化定义

Formally, a combinatorial optimization problem [math]\displaystyle{ A }[/math] is a quadruple[citation needed] [math]\displaystyle{ (I, f, m, g) }[/math], where

Formally, a combinatorial optimization problem [math]\displaystyle{ A }[/math] is a quadruple [math]\displaystyle{ (I, f, m, g) }[/math], where

从形式上来说,一个组合优化问题[math]\displaystyle{ A }[/math]是涉及四个变量[math]\displaystyle{ (I,f,m,g) }[/math]的问题 :


  • [math]\displaystyle{ I }[/math] is a set of instances;

[math]\displaystyle{ I }[/math]是实例中的数学集合;

  • given an instance [math]\displaystyle{ x \in I }[/math], [math]\displaystyle{ f(x) }[/math] is the finite set of feasible solutions;

给定[math]\displaystyle{ I }[/math]中的一个实例,[math]\displaystyle{ f(x) }[/math]是可行解的有限集合;

  • given an instance [math]\displaystyle{ x }[/math] and a feasible solution [math]\displaystyle{ y }[/math] of [math]\displaystyle{ x }[/math], [math]\displaystyle{ m(x, y) }[/math] denotes the measure of [math]\displaystyle{ y }[/math], which is usually a positive real.

给定一个实例[math]\displaystyle{ x }[/math]和其对应的可行解[math]\displaystyle{ y }[/math][math]\displaystyle{ m(x,y) }[/math]表示[math]\displaystyle{ y }[/math]的测度,其中,y通常是正实数。

  • [math]\displaystyle{ g }[/math] is the goal function, and is either [math]\displaystyle{ \min }[/math] or [math]\displaystyle{ \max }[/math].

[math]\displaystyle{ 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]\displaystyle{ x }[/math] an optimal solution, that is, a feasible solution [math]\displaystyle{ y }[/math]

The goal is then to find for some instance [math]\displaystyle{ x }[/math] an optimal solution, that is, a feasible solution [math]\displaystyle{ y }[/math]

然后,我们的目标是找到实例[math]\displaystyle{ x }[/math]的一个最优解,也就是可行解[math]\displaystyle{ y }[/math]



[math]\displaystyle{ \lt math\gt 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>



For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure [math]\displaystyle{ m_0 }[/math]. For example, if there is a graph [math]\displaystyle{ G }[/math] which contains vertices [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math], an optimization problem might be "find a path from [math]\displaystyle{ u }[/math] to [math]\displaystyle{ 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]\displaystyle{ u }[/math] to [math]\displaystyle{ 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]\displaystyle{ m_0 }[/math]. For example, if there is a graph [math]\displaystyle{ G }[/math] which contains vertices [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math], an optimization problem might be "find a path from [math]\displaystyle{ u }[/math] to [math]\displaystyle{ 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]\displaystyle{ u }[/math] to [math]\displaystyle{ 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,它查找是否存在某一特定测度的可行解。例如,如果一个图形 [math]\displaystyle{ G }[/math] 包含顶点 [math]\displaystyle{ u }[/math][math]\displaystyle{ v }[/math] ,那么最优化问题可能是“找到一条从[math]\displaystyle{ u }[/math][math]\displaystyle{ v }[/math]使用最少边的路径”。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]\displaystyle{ u }[/math][math]\displaystyle{ v }[/math] 使用10条或更少边的路径? ”这个问题可以用简单的“是”或“否”来回答。


In the field of approximation 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.[11]

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.

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。[11]

NP optimization problem NP优化问题

An NP-optimization problem (NPO) is a combinatorial optimization problem with the following additional conditions.[12] 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.

NP优化问题(NPO)是一个具有以下附加条件的组合优化问题。[12]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]\displaystyle{ y\in f(x) }[/math] is polynomially bounded in the size of the given instance [math]\displaystyle{ x }[/math],

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

  • the languages [math]\displaystyle{ \{\,x\,\mid\, x \in I \,\} }[/math] and [math]\displaystyle{ \{\,(x,y)\, \mid\, y \in f(x) \,\} }[/math] can be recognized in polynomial time, and

语言[math]\displaystyle{ \{\,x\,\mid\, x \in I \,\} }[/math] and [math]\displaystyle{ \{\,(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,

-- Flipped讨论) Languages 的理解


[math]\displaystyle{ 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.[13]

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。[13]


NPO is divided into the following subclasses according to their approximability:[12]

NPO is divided into the following subclasses according to their approximability:

NPO问题根据其近似性可分为以下子类: [12]


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 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(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]\displaystyle{ 1/c }[/math] of the optimal cost (for maximization problems). In 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 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]\displaystyle{ 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):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 TSP and 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


An NPO problem is called polynomially bounded (PB) if, for every instance [math]\displaystyle{ x }[/math] and for every solution [math]\displaystyle{ y\in f(x) }[/math], the measure [math]\displaystyle{ m(x, y) m(x, y) M (x,y) }[/math]is bounded by a polynomial function of the size of [math]\displaystyle{ x }[/math]. The class NPOPB is the class of NPO problems that are polynomially-bounded.


An NPO problem is called polynomially bounded (PB) if, for every instance [math]\displaystyle{ x }[/math] and for every solution [math]\displaystyle{ y\in f(x) }[/math], the measure is bounded by a polynomial function of the size of [math]\displaystyle{ x }[/math]. The class NPOPB is the class of NPO problems that are polynomially-bounded.


如果对于每个实例[math]\displaystyle{ x }[/math][math]\displaystyle{ f(x) }[/math]中的每个解[math]\displaystyle{ y }[/math],其测度[math]\displaystyle{ m(x,y) , M(x,y) }[/math]被一个大小为[math]\displaystyle{ 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。





Specific problems 特定问题

文件:TSP Deutschland 3.png
An optimal traveling salesperson tour through Germany’s 15 largest cities. It is the shortest among 43,589,145,600[14] 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.

最优的旅行商途经[[德国15个最大的城市。这是每个城市只能访问一次的43589145600[15]次旅行中最短的一次。]

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

See also

约束复合图

Notes

  1. 脚本错误:没有“Footnotes”这个模块。.
  2. 脚本错误:没有“Footnotes”这个模块。.
  3. Discrete Optimization. Elsevier. http://www.elsevier.com/locate/disopt. Retrieved 2009-06-08. 
  4. Discrete Optimization. Elsevier. http://www.elsevier.com/locate/disopt. Retrieved 2009-06-08. 
  5. Sbihi, Abdelkader; Eglese, Richard W. (2007). "Combinatorial optimization and Green Logistics" (PDF). 4Or. 5 (2): 99–116. doi:10.1007/s10288-007-0047-3.
  6. Sbihi, Abdelkader; Eglese, Richard W. (2007). "Combinatorial optimization and Green Logistics" (PDF). 4Or. 5 (2): 99–116. doi:10.1007/s10288-007-0047-3.
  7. Eskandarpour, Majid; Dejax, Pierre; Miemczyk, Joe; Péton, Olivier (2015). "Sustainable supply chain network design: An optimization-oriented review" (PDF). Omega. 54: 11–32. doi:10.1016/j.omega.2015.01.006.
  8. Eskandarpour, Majid; Dejax, Pierre; Miemczyk, Joe; Péton, Olivier (2015). "Sustainable supply chain network design: An optimization-oriented review" (PDF). Omega. 54: 11–32. doi:10.1016/j.omega.2015.01.006.
  9. 脚本错误:没有“Footnotes”这个模块。.
  10. 脚本错误:没有“Footnotes”这个模块。.
  11. 11.0 11.1 Ausiello, Giorgio; et al. (2003), Complexity and Approximation (Corrected ed.), Springer, ISBN 978-3-540-65431-5
  12. 12.0 12.1 12.2 12.3 Hromkovic, Juraj (2002), Algorithmics for Hard Problems, Texts in Theoretical Computer Science (2nd ed.), Springer, ISBN 978-3-540-44134-2
  13. 13.0 13.1 Kann, Viggo (1992), On the Approximability of NP-complete Optimization Problems, Royal Institute of Technology, Sweden, ISBN 91-7170-082-X
  14. 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.
  15. 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.


References

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第一个 = j. e。. [http://people.brunel.ac.uk/~mastjjb/jeb/or/ip.html

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类型 = 课堂笔记). line feed character in |title= at position 20 (help); line feed character in |first= at position 6 (help); line feed character in |ref= at position 5 (help); line feed character in |type= at position 14 (help); line feed character in |url= at position 51 (help)

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1-link = William j. Cook |Cook

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第一名: 威廉 (2016

2016年). [http://www.tsp.gatech.edu/optimal/index.html

Http://www.tsp.gatech.edu/optimal/index.html "Optimal TSP Tours"] Check |url= value (help). 滑铁卢大学. line feed character in |first= at position 8 (help); line feed character in |ref= at position 5 (help); line feed character in |url= at position 45 (help); line feed character in |year= at position 5 (help); Check date values in: |year= (help) (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)


  • Crescenzi

1 = Crescenzi, Pierluigi

1 = Pierluigi; Kann

2 = Kann, Viggo

2 = Viggo; Halldórsson, Magnús; [[Marek Karpinski

4-link = Marek Karpinski|Karpinski

4 = Karpinski, Marek

4 = Marek]]; [[Gerhard J. Woeginger

5-link = Gerhard j. Woeginger|Woeginger

5 = Woeginger, Gerhard]] (eds.). [http://www.nada.kth.se/%7Eviggo/wwwcompendium/

Http://www.nada.kth.se/%7eviggo/wwwcompendium/ "A Compendium of NP Optimization Problems 最优化问题概要"] Check |url= value (help). Unknown parameter |编辑器-first5= ignored (help); line feed character in |editor-last4= at position 10 (help); line feed character in |editor5-link= at position 21 (help); line feed character in |editor-last2= at position 5 (help); line feed character in |editor-first4= at position 6 (help); line feed character in |editor-first2= at position 6 (help); line feed character in |url= at position 47 (help); line feed character in |editor-first1= at position 10 (help); line feed character in |editor4-link= at position 16 (help); line feed character in |title= at position 41 (help); line feed character in |editor-last5= at position 10 (help); line feed character in |ref= at position 5 (help); line feed character in |editor-last1= at position 10 (help) (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NP 优化问题的近似结果的目录。)


  • Quantum Annealing and Related Optimization Methods

量子退火和相关的优化方法. Lecture Notes in Physics. '679

679. Springer. 2005

2005年. Bibcode [http://adsabs.harvard.edu/abs/2005qnro.book.....D

2005 qnro. book... d 2005qnro.book.....D 2005 qnro. book... d]. 

}}
}}


  • Das

1 = Das, Arnab

1 = Arnab; Chakrabarti

2 = Chakrabarti, Bikas K

2 = Bikas k (2008

2008年). "学术讨论会: 量子退火和模拟量子计算". Rev. Mod. Phys. 80

80 (3

第三期): 1061

1061. arXiv:[//arxiv.org/abs/0801.2193

0801.2193 0801.2193 0801.2193] Check |arxiv= value (help). Bibcode:[https://ui.adsabs.harvard.edu/abs/2008RvMP...80.1061D

2008/rvmp... 80.1061 d 2008RvMP...80.1061D 2008/rvmp... 80.1061 d] Check |bibcode= length (help). CiteSeerX [//citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.563.9990%0A%0A10.1.1.563.9990 10.1.1.563.9990 10.1.1.563.9990] Check |citeseerx= value (help). doi:10.1103/RevModPhys. 80.1061 Check |doi= value (help). Unknown parameter |日记= ignored (help); line feed character in |last2= at position 12 (help); line feed character in |issue= at position 2 (help); line feed character in |citeseerx= at position 16 (help); line feed character in |last1= at position 4 (help); line feed character in |first2= at position 8 (help); line feed character in |first1= at position 6 (help); line feed character in |arxiv= at position 10 (help); line feed character in |volume= at position 3 (help); line feed character in |page= at position 5 (help); line feed character in |bibcode= at position 20 (help); line feed character in |year= at position 5 (help); line feed character in |ref= at position 5 (help); Check date values in: |year= (help)

}}
}}


2001年). Combinatorial Optimization: Networks and Matroids

组合优化: 网络与拟阵. Dover. ISBN 0-486-41453-1. 

}}

}}


  • [[Jon Lee (mathematician)

乔恩 · 李(数学家) |Lee, Jon]] (2004

2004年). [https://books.google.com/books?id=3pL1B7WVYnAC

Https://books.google.com/books?id=3pl1b7wvynac A First Course in Combinatorial Optimization

组合优化的第一堂课]. Cambridge University Press

剑桥大学出版社. ISBN 0-521-01012-8. https://books.google.com/books?id=3pL1B7WVYnAC

Https://books.google.com/books?id=3pl1b7wvynac. 

}}

}}


  • Papadimitriou

1 = Papadimitriou, Christos H.

1 = Christos h.; Steiglitz 2 = Steiglitz, Kenneth 2 = Kenneth (July 1998

日期 = 1998年7月). Combinatorial Optimization : Algorithms and Complexity

组合优化: 算法与复杂性. Dover. ISBN 0-486-40258-4. 

}}

}}


  • Schrijver, Alexander

第一个 = 亚历山大 (2003

2003年). [https://books.google.com/books?id=mqGeSQ6dJycC

Https://books.google.com/books?id=mqgesq6djycc Combinatorial Optimization: Polyhedra and Efficiency

组合优化: 多面体与效率]. Algorithms and Combinatorics

序列 = 算法和组合数学. 24

24. Springer. ISBN [[Special:BookSources/9783540443896

9783540443896|9783540443896

9783540443896]]. https://books.google.com/books?id=mqGeSQ6dJycC

Https://books.google.com/books?id=mqgesq6djycc. 

}}
}}


  • Schrijver, Alexander

第一个 = 亚历山大 (2005

2005年). "On the history of combinatorial optimization (till 1960)". 离散优化手册. Elsevier. pp. 1–68. http://homepages.cwi.nl/~lex/files/histco.pdf. 

}}

}}


  • Schrijver, Alexander

第一个 = 亚历山大 (February 1, 2006

日期 = 2006年2月1日). [http://homepages.cwi.nl/~lex/files/dict.pdf

Http://homepages.cwi.nl/~lex/files/dict.pdf A Course in Combinatorial Optimization

文章标题: 组合优化课程]. http://homepages.cwi.nl/~lex/files/dict.pdf

Http://homepages.cwi.nl/~lex/files/dict.pdf. 

}}

}}


  • [[Gerard Sierksma

1-link = Gerard Sierksma |Sierksma

1 = Sierksma, Gerard

1 = Gerard]]; Ghosh

2 = Ghosh, Diptesh

2 = Diptesh (2010

2010年). 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. Springer. ISBN 978-1-4419-5512-8. 

}}

}}


  • Gerard Sierksma

1 = Gerard Sierksma; Yori Zwols

2 = Yori Zwols (2015

2015年). 线性和整数优化: 理论与实践. CRC Press. ISBN 978-1-498-71016-9. 

}}

}}


  • pinter, C-M. (2014

2014年). [https://www.springer.com/la/book/9783642401787

Https://www.springer.com/la/book/9783642401787 组合优化问题的仿生计算进展]. 智能系统参考库. Springer. ISBN 978-3-642-40178-7. https://www.springer.com/la/book/9783642401787

Https://www.springer.com/la/book/9783642401787. 

}}
}}


External links

模板:Commonscat

Category:Computational complexity theory

类别: 计算复杂性理论

Category:Theoretical computer science

类别: 理论计算机科学


eo:Diskreta optimumigo

eo:Diskreta optimumigo

2: Diskreta optiumigo


This page was moved from wikipedia:en:Combinatorial optimization. Its edit history can be viewed at 组合优化/edithistory