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Probabilistic Checking of Proofs: A New Characterization of NP \Lambda

Summary: Probabilistic Checking of Proofs: A New
Characterization of NP \Lambda
Sanjeev Arora y Shmuel Safra z
We give a new characterization of NP: the class NP contains exactly those languages
L for which membership proofs (a proof that an input x is in L) can be verified
probabilistically in polynomial time using logarithmic number of random bits and by
reading sub­logarithmic number of bits from the proof.
We discuss implications of this characterization; specifically, we show that approx­
imating Clique and Independent Set, even in a very weak sense, is NP­hard.
categories and subject descriptors: F.1.2 (Modes of Computation); F.1.3
(Complexity Classes); F.2.1 (Numerical Algorithms); F.2.2 (Nonnumerical Algorithms);
F.4.1 (Mathematical Logic).
1 Introduction
Problems involving combinatorial optimization arise naturally in many applications. For
many problems, no polynomial­time algorithms are known. The work of Cook, Karp, and
Levin [Coo71, Kar72, Lev73] provides a good reason why: many of these problems are NP­
hard. If they were to have polynomial­time algorithms, then so would every NP decision
problem, and so P = NP. Thus if P 6= NP --- as is widely believed --- then an NP­hard
problem has no polynomial­time algorithm.


Source: Arora, Sanjeev - Department of Computer Science, Princeton University


Collections: Computer Technologies and Information Sciences