The PCP theorem and hardness of approximation
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The PCP Theorem and Hardness of Approximation
In this report, we give a brief overview of Probabilistically Chekable proofs, and present Irit Dinur’s proof of the celebrated PCP theorem. We also briefly deal with the importance of the PCP theorem in deciding hardness of approximation of various problems.
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– PCP theorem. The classic PCP theorem is most useful for proving that optimization problems have no PTAS. – E3LIN2. The 1− 2 vs 1/2− 2 hardness of this problem is most useful for proving that problems have no approximation beyond a fixed constant factor; i.e., for statements like “such-and-such maximization problem has no 77 78 -approximation unless P = NP”. – Raz’s Label Cover. The hardness o...
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In this paper, we will first define basic concepts in computational complexity. Then, we will introduce the PCP Theorem and its equivalent formulation from the hardness of approximation view. Finally, we will prove a weaker version of the PCP theorem.
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This paper introduces basic notions of classic computational complexity theory and probabilistically checkable proof. It sketches the proof of the PCP theorem, and discusses its applications to the hardness of approximation of various problems.
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