On some relations between approximation problems and PCPs over the real numbers
Klaus Meer
Theory of Computing Systems 2005.

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Abstract

It was recently shown that $NP_{\mathbb{R}} \subseteq PCP_{\mathbb{R}}(poly, O(1)),$ i.e.the existence of transparent long proofs for $NP_{\mathbb{R}}$ was established. The latter denotes the class of real number decision problems verifiable in polynomial time as introduced by Blum, Shub and Smale. The present paper is devoted to the question what impact a potential full real number PCP$_{\mathbb{R}}$ theorem $NP_{\mathbb{R}} = PCP_{\mathbb{R}}(O(\log{n}),O(1))$ 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 $\mathbb{R}$ with approximation issues for these two problems. The first main result characterizes validity of a full PCP$_{\mathbb{R}}$ theorem by the existence of a certain reduction from MAX-QPS to MAX-q-CAP. The second result proves non-existence of particular approximation algorithms if we assume $NP_{\mathbb{R}} = PCP_{\mathbb{R}}(O(\log{n}),O(1))$ to hold.

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• On some relations between approximation problems and PCPs over the real numbers (deposited 09 April 2005) [Currently Displayed]
  • On some relations between approximation problems and PCPs over the real numbers (deposited 28 November 2005)

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