On Parallel versus Sequential Approximation

In this paper we deal with the class NCX of NP Optimization problems that are approximable within constant ratio in NC. This class is the parallel counterpart of the class APX. Our main motivation here is to reduce the study of sequential and parallel approximability to the same framework. To this aim, we rst introduce a new kind of NC-reduction that preserves the relative error of the approximate solutions and show that the class NCX has complete problems under this reducibility. An important subset of NCX is the class MAXSNP, we show that MAXSNP-complete problems have a threshold on the parallel approximation ratio that is, there are positive constants " 1 , " 2 such that although the problem can be approximated in P within " 1 it cannot be approximated in NC within " 2 , unless P=NC. This result is attained by showing that the problem of approximating the value obtained through a non-oblivious local search algorithm is P-complete, for some values of the approximation ratio. Finally, we show that approximating through non-oblivious local search is in average NC.