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 ofsearch algorithm ormetaheuristic can be used to solve them. Widely applicable approaches includebranch-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) andtabu 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 areNP-complete, such as the traveling salesman (decision) problem,[6] this is expected unlessP=NP.
For each combinatorial optimization problem, there is a correspondingdecision problem that asks whether there is a feasible solution for some particular measure. For example, if there is agraph 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 ofapproximation 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 referredpolynomials 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 polynomiallybounded in the size of the given instance,
the languages of valid instances and of valid instance–solution pairs can berecognized inpolynomial time, and
This implies that the corresponding decision problem is inNP. 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 areNP-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 usualTuring andKarp reductions. An example of such a reduction would beL-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(III): The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at mostc times the optimal cost (for minimization problems) or a cost at least of the optimal cost (for maximization problems). InHromkovič's bookAlgorithms for Hard Problems, excluded from this class are all NPO(II)-problems save if P=NP.[8] Without the exclusion, equals APX. ContainsMAX-SAT and metricTSP.
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 theset 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 theTSP andclique problem.
An NPO problem is calledpolynomially 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.
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An optimal traveling salesman tour throughGermany’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
^abcHromkovic, 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.
