Complexity issues on PTAS

Abstract

This chapter surveys various complexity issues of polynomial time approximation schemes (PTAS). Also under consideration are the refined subclasses of PTAS, including fully polynomial time approximation schemes (FPTAS) and efficient polynomial time approximation schemes (EPTAS). The key questions under consideration are as follows: What optimization problems have PTASs, FPTASs, and EPTASs, and how efficient are these algorithms? On the positive side, there are syntactic classes for characterizing PTAS, FPTAS, and EPTAS. Expressing a problem as a member in these classes automatically gives an approximation scheme for the problem. For many graph problems on planar graphs, there are powerful graph-theory techniques for designing PTAS and EPTAS algorithms, which can be extended to some generalizations of planar graphs. On the negative side, there are convincing evidences that, for a large set of NP-hard optimization problems, efficient approximation schemes do not exist. Even for many problems that have PTASs or EPTASs, their running time has a strong lower bound.