Aminimum spanning tree of a weighted planar graph. Finding a minimum spanning tree is a common problem involving combinatorial optimization.
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects,[1] where the set of feasible solutionsisdiscrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), 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.
solving real-world instances that arise in practice and do not necessarily exhibit the worst-case behavior of in NP-complete problems (e.g. real-world TSP instances with tens of thousands of nodes[5]).
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 algorithmormetaheuristic can be used to solve them. Widely applicable approaches include 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 (decision) problem,[6] 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.[7]
AnNP-optimization problem (NPO) is a combinatorial optimization problem with the following additional conditions.[8] 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 is polynomially bounded in the size of the given instance ,
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 L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.[9]
NPO is divided into the following subclasses according to their approximability:[8]
NPO(II): Equals PTAS. Contains the Makespan scheduling problem.
NPO(III): :The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at most c times the optimal cost (for minimization problems) or a cost at least of the optimal cost (for maximization problems). In Hromkovič's book[which?], excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains MAX-SAT and metric TSP.
NPO(IV): :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(III)-problems are excluded from this class unless P=NP. Contains the set cover problem.
NPO(V): :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(IV)-problems are excluded from this class unless P=NP. Contains the TSP and clique problem.
An NPO problem is called polynomially bounded (PB) 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.
An optimal traveling salesperson tour through Germany’s 15 largest cities. It is the shortest among the 43,589,145,600[10] possible tours that visit each city exactly once.
^Ausiello, Giorgio; et al. (2003), Complexity and Approximation (Corrected ed.), Springer, ISBN978-3-540-65431-5
^ abHromkovic, Juraj (2002), Algorithmics for Hard Problems, Texts in Theoretical Computer Science (2nd ed.), Springer, ISBN978-3-540-44134-2
^Kann, Viggo (1992), On the Approximability of NP-complete Optimization Problems, Royal Institute of Technology, Sweden, ISBN91-7170-082-X
^Take one city, and take all possible orders of the other 14 cities. Then divide by two because it does not matter in which direction in time they come after each other: 14!/2 = 43,589,145,600.
is polynomially bounded in the size of the given instance 
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and 
can be recognizedinpolynomial time, and
ispolynomial-time computable.
of the optimal cost (for maximization problems). In Hromkovič's book[which?], excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains MAX-SAT and metric TSP.
