Polynomial Time Approximation Schemes

Abstract

Let Π be an NP-hard optimization problem, and let A be an approximation algorithm for Π. For an instance I of Π, let A(I) denote the objective value when running A on I, and let OPT (I) denote the optimal objective value. The approximation ratio of A for the instance I is RA(I) = A(I)/OPT (I), thus, when Π is minimization (maximization) problem RA(I) ≥ 1 (RA(I) ≤ 1). A polynomial time approximation scheme is an algorithm which takes as input an additional parameter, e, which determines the desired approximation ratio. This ratio can be arbitrarily close to 1, when e approaches 0. The time complexity of the scheme is polynomial in the input size but may be exponential in 1/e. This gives a clear trade-off between running time and quality of approximation. Formally,