Some Relations between Approximation Problems and PCPs over the Real Numbers

Abstract

In [10] it was recently shown that $\mbox{\rm NP}_{\Bbb R} \subseteq \mbox{\rm PCP}_{\Bbb R}(\,{\it poly}, O(1)),$ that is the existence of transparent long proofs for $\mbox{\rm NP}_{\Bbb R}$ was established. The latter denotes the class of real number decision problems verifiable in polynomial time as introduced by Blum et al. [6]. The present paper is devoted to the question what impact a potential full real number $\mbox{\rm PCP}_{\Bbb R}$ theorem $\mbox{\rm NP}_{\Bbb R} = \mbox{\rm PCP}_{\Bbb R}(O(\log{n}), O(1))$ would have on approximation issues in the BSS model of computation. We study two natural optimization problems in the BSS model. The first, denoted by MAX-QPS, is related to polynomial systems; the other, MAX-q-CAP, deals with algebraic circuits. Our main results combine the PCP framework over ${\Bbb R}$ with approximation issues for these two problems. We also give a negative approximation result for a variant of the MAX-QPS problem.