Almost transparent short proofs for NPR
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
Approximation classes for real number optimization problems
UC'06 Proceedings of the 5th international conference on Unconventional Computation
Optimization and approximation problems related to polynomial system solving
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
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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.