Combinatorial optimization

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 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.
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.
Applications.
Applications for combinatorial optimization include, but are not limited to:
Methods.
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:
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.
Formal definition.
Formally, a combinatorial optimization problem formula_1 is a quadruple formula_2, where
The goal is then to find for some instance formula_6 an "optimal solution", that is, a feasible solution formula_7 with
For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure formula_17. For example, if there is a graph formula_18 which contains vertices formula_19 and formula_20, an optimization problem might be "find a path from formula_19 to formula_20 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 formula_19 to formula_20 that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.
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.
NP optimization problem.
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.
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.
NPO is divided into the following subclasses according to their approximability:
An NPO problem is called "polynomially bounded" (PB) if, for every instance formula_6 and for every solution formula_25, the measure formula_33is bounded by a polynomial function of the size of formula_6. The class NPOPB is the class of NPO problems that are polynomially-bounded.