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.
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Meer, K. Some Relations between Approximation Problems and PCPs over the Real Numbers. Theory Comput Syst 41, 107–118 (2007). https://doi.org/10.1007/s00224-006-1336-5
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DOI: https://doi.org/10.1007/s00224-006-1336-5