组合优化

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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 。找到一个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.

组合优化是最优化 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。

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 < ! -- 同义词还是子域？: 离散优化 Discrete Optimization -- > 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 Traveling Salesman Problem (“ TSP”)、最小生成树问题(“ MST”)和背包问题 Knapsack Problem 。

Some research literature^[2] 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.

一些研究文献认为离散优化是由整数规划 Integer Programming 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

应用

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

组合优化的应用包括但不限于:

Logistics^[3]

物流

Supply chain optimization^[4]

供应链优化

Developing the best airline network of spokes and destinations

发展最好的航空公司网络的辐条和目的地

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

决定车队中哪辆出租车的路线来收取费用

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

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.

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 Polynomial Algorithm 算法的，其中相当一部分是由线性规划 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:

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)

多项式时间精确可解手头问题的特殊情况（例如，见（固定参数可处理））

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，旅行商问题#平方中随机点集的TSP路径长度TSP）

approximation algorithms 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”最优解的

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^[5]).

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)

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 Branch-and-bound (一种可以在任何时间点停止作为启发式算法的精确算法)、分支切面法 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 (一种贪婪型交换算法)。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]是一个关于四变量的[math]\displaystyle{ (I，f，m，g) }[/math]问题：

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

[math]\displaystyle{ I }[/math]是实例的数学中的集合；

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]和一个可行解[math]\displaystyle{ y }[/math]，[math]\displaystyle{ m(x,y) }[/math]表示[math]\displaystyle{ y }[/math]的映射函数。 --该句存疑

[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，可以是求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] with

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] with

然后，我们的目标是找到一个最优解[math]\displaystyle{ x }[/math]，也就是一个可行的解[math]\displaystyle{ y }[/math]

[math]\displaystyle{ \lt math\gt 《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]\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]\displaystyle{ G }[/math] 包含顶点 [math]\displaystyle{ u }[/math] 和 [math]\displaystyle{ v }[/math] ，那么一个最佳化问题是“找到一条从[math]\displaystyle{ u }[/math]到[math]\displaystyle{ v }[/math]且使用最少边的路径”。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] 到 [math]\displaystyle{ 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 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.^[6]

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优化问题。

NP optimization problem

NP优化问题 NP Optimization Problem

An NP-optimization problem (NPO) is a combinatorial optimization problem with the following additional conditions.^[7] 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。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的大小。

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],

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

[math]\displaystyle{ m }[/math] is polynomial-time computable.

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.^[8]

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:^[7]

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 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]\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(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 TSP and 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{ An NPO problem is called polynomially bounded (PB) if, for every instance \lt math\gt x }[/math] and for every solution [math]\displaystyle{ y\in f(x) }[/math], the measure [math]\displaystyle{ 一个 NPO 问题被称为多项式有界(PB) ，如果对于每个实例 \lt math \gt x }[/math] 和 f (x) </math > 中的每个解 < math > y，度量值 < math >

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.

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/>

Specific problems

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^[9] 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

Notes

↑ Schrijver 2003, p. 1.
↑ Discrete Optimization. Elsevier. http://www.elsevier.com/locate/disopt. Retrieved 2009-06-08.
↑ Sbihi, Abdelkader; Eglese, Richard W. (2007). "Combinatorial optimization and Green Logistics" (PDF). 4Or. 5 (2): 99–116. doi:10.1007/s10288-007-0047-3.
↑ 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.
↑ Cook 2016.
↑ Ausiello, Giorgio; et al. (2003), Complexity and Approximation (Corrected ed.), Springer, ISBN 978-3-540-65431-5
↑ ^7.0 ^7.1 Hromkovic, Juraj (2002), Algorithmics for Hard Problems, Texts in Theoretical Computer Science (2nd ed.), Springer, ISBN 978-3-540-44134-2
↑ Kann, Viggo (1992), On the Approximability of NP-complete Optimization Problems, Royal Institute of Technology, Sweden, ISBN 91-7170-082-X
↑ 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.

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