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 oflinear 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:
polynomial-time exactly solvable special cases of the problem at hand
algorithms that perform well on "random" instances
solving real-world instances that arise in practice and do not necessarily exhibit the worst-case behavior inherent 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. Perhaps the most universally applicable approaches are 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.
The goal is then to find for some instance an optimal solution, that is, a feasible solution with 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'. 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.
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 is polynomially bounded in the size of the given instance,
the languages and can be recognized in polynomial time, and
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 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:
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, 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 Hromkovic'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 Max Clique problems.
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.
