组合最优化 Combinatorial optimization 又称组合规划，在实践中有着广泛的应用，同时也是数学（运筹学，组合优化及图论）和计算机科学中的重要研究课题。它是在给定有限集的所有具备某些特性的子集中，按某种目标找出一个最优的子集的一类数学规划。初期,它所研究的问题,如广播网的设置、开关电路设计、航船运输路线的计划、旅游路线的安排、人与工作间的对应分配、课程时间表的制定、装箱（装车）问题、运输网络中或工程施工中薄弱环节的分析、制定露天煤矿开采计划等，都是网络上的一些极值问题，后来，对这些问题进行概括和抽象，在理论上研究了拟阵中的一些更一般性的组合最优化问题及其算法。

目前， 图论与组合优化理论与算法/技术几乎应用到工农业生产，科学研究，生活的所有方面，除了上面所述以外，还包括现今出现的各种新技术新行业， 例如：

• 能源组合优化
• 输电线路的优化
• 多种运输方式的组合优化
• 公路网布局优化
• 产品组合优化
• 销售和资产组合优化
• 投资组合优化
• 贷款组合优化
• Web服务组合优化方法
• 个人学业， 职业，健康保健，金融理财等方面的最优规划
  等等

图论中有一个有趣的结论， 就是1960年代发现的“小世界定律 Small World Theorem”。 根据该定律， 世界上任何二个人都可以通过不超过另外5个人建立联系。例如本人只需要通过其中二个人就可以跟欧洲某皇室建立联系， 如果有必要的话。 但对其他人而言， 如何在所需要的人之间建立一种联系， 是需要通过优化设计的并付出相当代价的。

总而言之， 凡是需要找到最优，最佳方案（最大， 最小，最长，最短等等）的地方几乎都可以通过组合优化方式寻得解决办法。 当然， 并非所有问题都能找到最优方案， 但许多情况下可以逼近最佳方案。

需要这方面解决方案的单位与个人可与本人联系。

本人简历：

本人国内某高校硕士，国外几所大学博士资格获得者，在国内外若干著名公司企业服务多年，会英法德俄等语。另外本人自主创业，从事各类咨询多年。

Graph Theory, Combinatorial Optimization and Their Applications for You

Combinatorics, the mathematics of finite (or countable) structures is often used in computer science and communications. Optimization explores ways to make any operation work more efficiently within given constraints.

Combinatorics is, in essence, the study of arrangements: pairings and groupings, rankings and orderings, selections and allocations. There are three principal branches in the subject. Enumerative combinatorics is the science of counting. Problems in this subject deal with determining the number of possible arrangements
of a set of objects under some particular constraints. Existential combinatorics studies problems concerning the existence of arrangements that possess some specified property. Constructive combinatorics is the design and study of algorithms for creating arrangements with special properties.

In applied mathematics and theoretical computer science, 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 feasible. 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. Some common problems involving combinatorial optimization are the traveling salesman problem ("TSP") and the minimum spanning tree problem ("MST").

Combinatorial optimization is a subset 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, mathematics, auction theory, and software engineering.

Combinatorial optimization is an active field leveraging ideas from many different areas including graph theory, combinatorics, matroid theory, submodularity, connectivity, network flows, approximation algorithms, mathematical programming, game theory, algebraic and geometric methods, and applications.

Combinatorics is closely related to the theory of graphs. Many problems in graph theory concern arrangements of objects and so may be considered as combinatorial problems. For example, the theory of matchings and Ramsey theory, have the flavor of existential combinatorics. Also, combinatorial techniques are often employed to address problems in graph theory.

Methods & Algorithms

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. However, generic search algorithms are not guaranteed to find an optimal solution, 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.

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.

For NP-complete discrete optimization problems, current research literature includes the following topics:

• 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 TSP)
• approximation algorithms 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).

Applications

The formal study of combinatorics has seen a huge growth in the subject, fueled by problems and applications from many fields of study. Combinatorics and optimization give you powerful methods for modelling and solving large scheduling and management problems, in fields such as

• Chemistry, in studying arrangements of atoms in molecules and crystals;
• Biology, in questions about the structure of genes and proteins;
• Physics, in problems in statistical mechanics;
• Communications, in the design of codes for encryption and cryptography,
• Compression, and correction of errors;
• and especially computer science, for instance in problems of scheduling and allocating resources, and in analyzing the efficiency of algorithms.

The optimization problems in the above areas are more or less related to the following:

• Assignment problem
• Closure problem
• Constraint satisfaction problem
• Cutting stock problem
• Integer programming
• Knapsack problem
• Minimum spanning tree
• Traveling salesman problem

in the studies of graph theory and combinatorial optimization.

Specifically, they can deal with all optimization tasks encountered in everyday life, business, and researches, such as but not limited to:

• Concept testing
• Electronic security
• Equipment maintenance planning
• Factory floor layout
• Flight schedules / developing the best airline network of spokes and destinations
• Nurse scheduling
• Package delivery
• Staff scheduling
• telecommunications signal improvement
• Taxis fleet routing
• Traffic lights synchronizing
• Vehicle routing
• Vehicle rescheduling
• Weapon target assignment