On parallel versus sequential approximation

Abstract

Here we deal with the class NCX of optimization problems that arc approximable within constant ratio in NC. We first introduce a new kind of reduction that preserves the relative error of the approximate solutions and show that the class NCX has complete problems for this reducibility. An important subset of NCX is the class Max SNP, we show that Max SNP complete problems have a threshold on the parallel approximation ratio that is, there are constants ε₁, ε₂ such that although the problem can be approximated in P within ε₂ 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. We finally show that approximating within ∈ using non-oblivious local search is in average NC.

This research was supported by the ESPRIT BRA Program of the EC under contract no. 7141, project ALCOM II.