Question
Asked 10th Jan, 2015

# What is the meaning of combinatorial optimization?

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

## Most recent answer

7th Mar, 2021
Cenk Çalışkan
Utah Valley University
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.

## Popular answers (1)

15th Jan, 2015
Adam N. Letchford
Lancaster University
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).
17 Recommendations

## All Answers (16)

10th Jan, 2015
Humyun Fuad Rahman
Cardiff Met University
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
10 Recommendations
11th Jan, 2015
Daniel R. Page
PageWizard Games Learning & Entertainment
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.
4 Recommendations
12th Jan, 2015
Michael Patriksson
Chalmers University of Technology
"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.
5 Recommendations
13th Jan, 2015
Behnam Mohammadi-ivatloo
University of Tabriz
it is something like discrete optimization!
1 Recommendation
14th Jan, 2015
Nishant Kumar
Indian Institute of Technology Jodhpur
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).
2 Recommendations
15th Jan, 2015
Sarit Chakraborty
Government College of Engineering and Leather Technology
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" .........
4 Recommendations
15th Jan, 2015
Adam N. Letchford
Lancaster University
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).
17 Recommendations
16th Jan, 2015
Daniel R. Page
PageWizard Games Learning & Entertainment
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 ?"
2 Recommendations
16th Jan, 2015
Alireza Soroudi
The Institution of Engineering and Technology
Thank you Daniel and other researchers who have given me very interesting answers.
1 Recommendation
16th Jan, 2015
Imène Benkalai
University of Québec in Chicoutimi
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:
Best regards,
Imène.
5 Recommendations
18th Jan, 2015
Nizar Harb
Higher Colleges of Technology
simple examples of combinatorial problems that were intensively studied
- The traveling salesperson.
- black box optimization problem with five on-off switches
- Solving a maze.
3 Recommendations
20th Jul, 2017
Adam N. Letchford
Lancaster University
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.
13th Apr, 2019
Negar Sa
Amirkabir University of Technology
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.
1 Recommendation
7th Mar, 2021
Cenk Çalışkan
Utah Valley University
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.

## Related Publications

Article
DavisMartin D. and WeyukerElaine J.. Comparability, complexity, and languages. Fundamentals of theoretical computer science. Computer science and applied mathematics. Academic Press, New York etc. 1983, xix + 425 pp. - Volume 52 Issue 1 - Wolfgang Maass
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