Efficient approximation schemes for geometric problems?

Abstract

An EPTAS (efficient PTAS) is an approximation scheme where ε does not appear in the exponent of n, i.e., the running time is f(ε) · n c. We use parameterized complexity to investigate the possibility of improving the known approximation schemes for certain geometric problems to EPTAS. Answering an open question of Alber and Fiala [2], we show that MAXIMUM INDEPENDENT SET is W[1]-complete for the intersection graphs of unit disks and axis-parallel unit squares in the plane. A standard consequence of this result is that the nO(1/ε) time PTAS of Hunt et al. [11] for MAXIMUM INDEPENDENT SET on unit disk graphs cannot be improved to an EPTAS. Similar results are obtained for the problem of covering points with squares. © Springer-Verlag Berlin Heidelberg 2005.