APX

In computational complexity theory, the class APX (an abbreviation of "approximable") is the set of NP optimization problems that allow polynomial-time approximation algorithms with approximation ratio bounded by a constant (or constant-factor approximation algorithms for short). In simple terms, problems in this class have efficient algorithms that can find an answer within some fixed multiplicative factor of the optimal answer.

Definition

An approximation algorithm is called an

f

(

n

)

{\displaystyle f(n)}

-approximation algorithm for input size

n

{\displaystyle n}

if it can be proven that the solution that the algorithm finds is at most a multiplicative factor of

f

(

n

)

{\displaystyle f(n)}

times worse than the optimal solution. Here,

f

(

n

)

{\displaystyle f(n)}

is called the approximation ratio. Problems in APX are those with algorithms for which the approximation ratio

f

(

n

)

{\displaystyle f(n)}

is a constant

c

{\displaystyle c}

. The approximation ratio is conventionally stated greater than 1. In the case of minimization problems,

f

(

n

)

{\displaystyle f(n)}

is the found solution's score divided by the optimum solution's score, while for maximization problems the reverse is the case. For maximization problems, where an inferior solution has a smaller score,

f

(

n

)

{\displaystyle f(n)}

is sometimes stated as less than 1; in such cases, the reciprocal of

f

(

n

)

{\displaystyle f(n)}

is the ratio of the score of the found solution to the score of the optimum solution.

A problem is said to have a polynomial-time approximation scheme (PTAS) if for every multiplicative factor of the optimum worse than 1 there is a polynomial-time algorithm to solve the problem to within that factor. Unless P = NP there exist problems that are in APX but without a PTAS, so the class of problems with a PTAS is strictly contained in APX. One example of a problem with a PTAS is the knapsack problem.