Optimization problem: Difference between revisions

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Revision as of 04:09, 15 March 2010

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The goal is then to find for some instance <math>x</math> an ''optimal solution'', that is, a feasible solution <math>y</math> with

<center>: <math>

m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .

</math></center>

For each optimization problem, there is a corresponding [[decision problem]] that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a [[Graph (mathematics)|graph]] <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> 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 <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

In the field of [[approximation algorithmsalgorithm]]s, 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.<ref name=Ausiello03>{{citation

| last1 = Ausiello | first1 = G.

| last2 = et al.