Combinatorial optimization

Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem, the minimum spanning tree problem, and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.
Combinatorial optimization 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, VLSI, applied mathematics and theoretical computer science.

Applications

Basic applications of combinatorial optimization include, but are not limited to:

Logistics
Supply chain optimization
Developing the best airline network of spokes and destinations
Deciding which taxis in a fleet to route to pick up fares
Determining the optimal way to deliver packages
Allocating jobs to people optimally
Designing water distribution networks
Earth science problems

Methods

There is a large amount of literature on polynomial-time algorithms for certain special classes of discrete optimization. A considerable amount of it is unified by the theory of linear programming. Some examples of combinatorial optimization problems that are covered by 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
algorithms that perform well on "random" instances
approximation algorithms that run in polynomial time and find a solution that is close to optimal
parameterized approximation algorithms that run in FPT time and find a solution close to the optimum
solving real-world instances that arise in practice and do not necessarily exhibit the worst-case behavior of in NP-complete problems.

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. Widely applicable approaches include branch-and-bound, branch-and-cut, dynamic programming and tabu search. However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly. Since some discrete optimization problems are NP-complete, such as the traveling salesman problem, this is expected unless P=NP.
For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure. For example, if there is a graph which contains vertices and, an optimization problem might be "find a path from to 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 to that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.
The field of approximation algorithms deals with algorithms 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 then more naturally characterized as an optimization problem.

NP optimization problem

An NP-optimization problem 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.

the size of every feasible solution, where denotes the set of feasible solutions to instance, is polynomially bounded in the size of the given instance,
the languages of valid instances and of valid instance–solution pairs can be recognized in polynomial time, and
The measure of a solution to problem 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 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 L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.
NPO is divided into the following subclasses according to their approximability:NPO: Equals FPTAS. Contains the Knapsack problem.NPO: Equals PTAS. Contains the Makespan scheduling problem.NPO: The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at most c times the optimal cost or a cost at least of the optimal cost. In Hromkovič's book Algorithms for Hard Problems, excluded from this class are all NPO-problems save if P=NP. Without the exclusion, equals APX. Contains MAX-SAT and metric TSP.NPO: 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 Hromkovič's book, all NPO-problems are excluded from this class unless P=NP. Contains the set cover problem.NPO: 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-problems are excluded from this class unless P=NP. Contains the TSP and clique problem.
An NPO problem is called polynomially bounded if, for every instance and for every solution, the measure is bounded by a polynomial function of the size of. The class NPOPB is the class of NPO problems that are polynomially-bounded.

Specific problems

Image:TSP Deutschland 3.png|thumb|An optimal traveling salesman tour through Germany’s 15 largest cities. It is the shortest among the 43,589,145,600 possible tours that visit each city exactly once.