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).
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.