Optimization problem

In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. More formally, an optimization problem ${\displaystyle A}$ is a quadruple ${\displaystyle (I,f,m,g)}$, where

• ${\displaystyle I}$ is a set of instances;
• given an instance ${\displaystyle x\in I}$, ${\displaystyle f(x)}$ is the set of feasible solutions;
• given an instance ${\displaystyle x}$ and a feasible solution ${\displaystyle y}$ of ${\displaystyle x}$, ${\displaystyle m(x,y)}$ denotes the measure of ${\displaystyle y}$, which is usually a positive real.
• ${\displaystyle g}$ is the goal function, and is either ${\displaystyle \min }$ or ${\displaystyle \max }$.

The goal is then to find for some instance ${\displaystyle x}$ an optimal solution, that is, a feasible solution ${\displaystyle y}$ with

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

For each optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure ${\displaystyle m_{0}}$. For example, if there is a graph ${\displaystyle G}$ which contains vertices ${\displaystyle u}$ and ${\displaystyle v}$, an optimization problem might be "find a path from ${\displaystyle u}$ to ${\displaystyle v}$ 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 ${\displaystyle u}$ to ${\displaystyle v}$ 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.^[1]

NP optimization problems

An NP-optimization problem (NPO) is an optimization problem with the following additional conditions.^[2]

• the size of every feasible solution is polynomially bounded,
• the languages ${\displaystyle I}$ and ${\displaystyle f(x)}$ can be recognized in polynomial time, and
• m is polynomial-time computable.

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-hard. 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. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.^[3]

Notes

1. Ausiello, G.; et al. (2003), Complexity and Approximation, Springer, ISBN ISBN-13 978-3540654315 {{citation}}: Check |isbn= value: invalid character (help); Explicit use of et al. in: |last2= (help); Unknown parameter |Edition= ignored (|edition= suggested) (help)
2. Hromkovic, Juraj (2002), Algorithms for Hard Problems, Texts in Theoretical Computer Science, Springer, ISBN ISBN-13 978-3540441342 {{citation}}: Check |isbn= value: invalid character (help); Unknown parameter |Edition= ignored (|edition= suggested) (help)
3. Kann, Viggo (1992), On the Approximability of NP-complete Optimization Problems, Texts in Theoretical Computer Science, Royal Institute of Technology, Sweden, ISBN 91-7170-082-X

See also

• Optimization (mathematics)