Combinatorial Optimization Problem
Formally, a combinatorial optimization problem is a quadruple, where
- is a set of instances;
- given an instance, is the set of feasible solutions;
- given an instance and a feasible solution of, denotes the measure of, which is usually a positive real.
- is the goal function, and is either or .
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.
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