What is the meaning of combinatorial optimization?

Combinatorial optimization problem is an optimization problem, where an optimal solution has to be identified from a finite set of solutions. The solutions are normally discrete or can be formed into discrete. This is an important topic studied in operations research, software engineering, artificial intelligence, machine learning, and so on. Travelling sales man problem is one of the popular combinatorial optimization problem. You may see this link: http://homepages.cwi.nl/~lex/files/dict.pdf

See page 1 of Schrijver, Alexander. Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics 24. Springer (a standard text in this field of Theoretical Computer Science).  I like the way this book describes this field of Theoretical Computer Science and Applied Mathematics (which has applications in fields such as operations research, machine learning and so on).  Schrijver keeps it simple in the introduction, which is greatly appreciated from a researcher who has done so much in Combinatorial Optimization.

"Combinatorial" refers to an ordering of items. For example, the combinatorial optimization problem known as the traveling salesperson problem calls for an optimal ordering of the cities to be visited, such that the total length of travel is at a minimum. 

Some of these problems are very hard, like the TSP, while others are very simple. See 

for a discussion on one of the simplest combinatorial problems, namely the minimum spanning tree problem, as well as a few others.

it is something like discrete optimization!

Two of the best known, conceptually simple and computationally easy

combinatorial optimization problems are: to find the shortest path from a node s

to a node t in a directed graph with nonnegative edge lengths; and to find a

minimum spanning tree in a graph (more generally, a minimum rooted spanning

arborescence in a directed graph).

The term "Combinatorial" can be understood as  "combination of steps chosen from a series/ (set of steps) which will give the optimum result." How to chose such combination of steps or what mathematics to use is depend on the researcher.  it may be the optimum distance as far as  TSP is concerned (as correctly Refereed by Michael Patriksson ) but it's doubtful to say combinatorial optimization is exactly same as  "discrete optimization" .........

I always think of combinatorial optimization problems as being optimization problems in which the feasible solutions can be expressed using concepts from combinatorics (such as sets, subsets, combinations or permutations) and/or graph theory (such as vertices, edges, cliques, paths, cycles or cuts).

Most combinatorial optimization problems can be formulated as 0-1 Linear Programs, i.e., Linear Programs with the additional restriction that the variables must take binary values.  Then, to me, combinatorial optimization is a special case of discrete optimization (which just means optimizing over a discrete set --- not necessarily one with binary variables).

I think there are lovely answers here, but the question is asking for reports, papers and books that describe it:

"What is the meaning of combinatorial optimization?

Is there a simple (I mean simple) report or paper that describes it ?"

Thank you Daniel and other researchers who have given me very interesting answers.

I do not quite agree with the "ordering" concept, Mr. Patriksson. Some cominatorial optimization problems are about finding the optimal combination, set, or subset, etc. I may cite for example the knapsack problem. Not all combinatorial optimization problems deal with permutation problems like the TSP.

And to answer your question, Mr. Soroudi, I would advice you to read the following article:

http://www.math.uwaterloo.ca/~bico/papers/cobook_select.pdf (especially pages 6 and 13).

Best regards,

Imène.

simple examples of combinatorial problems that were intensively studied

- The traveling salesperson.

- black box optimization problem with five on-off switches

- Solving a maze.

The assignment problem is regarded as an (easy) combinatorial optimization problem. It is equivalent to finding a minimum cost perfect matching in a bipartite graph.

The problem of finding the optimal topological structure of computer networks in the sense of constructing them with maximum indices of their performance quality is a problem that uses discrete optimization (otherwise called combinatorial optimization) techniques over graphs. This paper attempts to solve this applied problem as an additional one besides those ones mentioned here in this very interesting discussion.

There is a good definition in " Blum C, Roli A. Metaheuristics in combinatorial optimization: Overview and conceptual comparison. ACM computing surveys (CSUR). 2003 Sep 1;35(3):268-308. " article :

Many optimization problems of practical

as well as theoretical importance consist

of the search for a “best” configuration

of a set of variables to achieve

some goals. They seem to divide naturally

into two categories: those where solutions

are encoded with real-valued variables,

and those where solutions are encoded

with discrete variables. Among the latter

ones we find a class of problems

called Combinatorial Optimization (CO)

problems. According to Papadimitriou and

Steiglitz, in CO problems, we are

looking for an object from a finite—or possibly

countably infinite—set. This object

is typically an integer number, a subset, a

permutation, or a graph structure.

Combinatorial optimization problems are problems with discrete and finite feasible spaces, where you need to optimize some objective function defined on that feasible space. Some typical examples are the travelling salesman problem (TSP), assignment problem, minimum spanning tree problem (MST) and the knapsack problem.