Theory of Computing Systems

, Volume 41, Issue 1, pp 107–118 | Cite as

Some Relations between Approximation Problems and PCPs over the Real Numbers

  • Klaus MeerEmail author


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.


Polynomial Time Approximation Problem Maximization Problem Polynomial System Common Zero 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Syddansk Universitet, Campusvej 555230 Odense MDenmark

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