Shmuel Safra

Probabilistic checking of proofs: a new characterization of NP

We give a new characterization of NP: the class NP contains exactly those languages <italic>L</italic> for which membership proofs (a proof that an input <italic>x</italic> is in <italic>L</italic>) can be verified probabilistically in polynomial time using <italic>logarithmic</italic> number of random bits and by reading <italic>sublogarithmic</italic> number of bits from the proof. We discuss implications of this characterization; specifically, we show that approximating Clique and Independent Set, even in a very weak sense, is NP-hard.

A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP

We introduce a new low-degree-test, one that uses the restriction of low-degree polynomials to planes (i. e., afine sub-spaces of dimension 2), rather than the restriction to lines (i. e., afine sub-spaces of dimension 1). We prove the new test to be of a very small emorprobability (in particular, much smaller than constant). The new test enables us to prove a low-error characterization of NP in terms of PCP. Specifically, OUT theorem states that, for any given c > 0, membership in any NP language can be verijied with 0(1) accesses, each r’eading logarithmic number of bits, and such that the error-probability is 2‘“~’-’ n. Our results are in fact stronger, as stated below. One application of the new characterization of NP is that approximating SETCOVER to within a logarithmic factors is NP-hard. Previous analysis for low-degree-tests, as well as previous characten”zations of NP in terms of PCP, have managed to achieve, with constant number of accesses, error-probability of, at best, a constant. The proof for the smail err-or-probability of our new low-degree-test is, nevertheless, significantly simpler than previous proofs. In particular, it is combinatorial and geometrical in nature, rather than algebraic. 1 Characterizations of NP in terms of PCP Characterizing the class NP, by itself or with respect to other computational-complexity clssses, is perhaps one *URL http: //www.math.tau. ac.il/school/courses/PCP tWejzmm ~st., ISRAEL. ~-a@wj~dom.wej--

Interactive proofs and the hardness of approximating cliques

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Probabilistic checking of proofs; a new characterization of NP

3.50 of the most fundamental avenues of research in theory of computer-science. Since the early days, when the classes P and NP were defined, and the question was posed as to whether they are the same or do they differ, many problems were shown to be NP-complete, thereby increasing the weight on finding stricter characterization for the class NP. NP has since been given a few alternative characterizations. The one most commonly applied being Cook’s [CO071], which characterizes NP in terms of efficient verification of proofs (or nondeterministic computations). A new perspective, by which improved characterizations of NP can be obtained, has been recently proposed. The motivation for which stems from questions regarding the hardness of approximation versions, for problems whose exact computation is known to be NP-hard. This avenue of research was initiated by [FGL+91], which introduced a new methodology for proving hardness results for approximation problems. The method takes advantage of results in a seemingly unrelated area — that of interactive proofs [GMR89, Bab85, BGKW88, LFKN92, Sha92, BFL91] — however interprets those results with quite a different perspective in mind. Much effort has been invested since towards a better understanding of this methodology, and the class NP has consequently gained stricter characterizations [AS92, ALM+92, BGS95], which are referred to as characterizations of NP in terms of PCP (or, in short, PCP characterizations of NP). The PCP characterization of NP — though has taken around 20 years to be formulated — seems now as the most natural extension of the old characterization of NP [CO071], if one has in mind proving hardness results for approximation problems. This characterization has already been used to obtain quite a few hardness results for approximation problems [FGL+91, AS92, ALM+ 92, PY91, LY94, BGLR93, KLS93, BGS95, Hiis96a, H5a96b, H5s97]. The previous characterization of NP in terms of the PCP hierarchy [AS92, ALM+ 92], seemed at first ss the best possible up to constant factors. A stronger characterization was later conjectured in [BGLR93]; one that, as an outcome, implies NP-hardness of approximating SET-COVER to within logarithmic factors [LY94, BGLR93]. The conjecture itself, more-

Algorithmic construction of sets for k -restrictions

The contribution of this paper is two-fold. First, a connection is established between approximating the size of the largest clique in a graph and multi-prover interactive proofs. Second, an efficient multi-prover interactive proof for NP languages is constructed, where the verifier uses very few random bits and communication bits. Last, the connection between cliques and efficient multi-prover interaction proofs, is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph. Of independent interest is our proof of correctness for the multilinearity test of functions.

The importance of being biased

The authors 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 sub-logarithmic number of queries to the proof. This is a non-relativizing characterization of NP. They discuss implications of this characterization; specifically, they show that approximating clique (or independent set) is NP-hard.<<ETX>>

Approximating-CVP to within almost-polynomial factors is NP-hard

This work addresses k-restriction problems, which unify combinatorial problems of the following type: The goal is to construct a short list of strings in Σm that satisfies a given set of k-wise demands. For every k positions and every demand, there must be at least one string in the list that satisfies the demand at these positions. Problems of this form frequently arise in different fields in Computer Science.The standard approach for deterministically solving such problems is via almost k-wise independence or k-wise approximations for other distributions. We offer a generic algorithmic method that yields considerably smaller constructions. To this end, we generalize a previous work of Naor et al. [1995]. Among other results, we enhance the combinatorial objects in the heart of their method, called splitters, and construct multi-way splitters, using a new discrete version of the topological Necklace Splitting Theorem [Alon 1987].We utilize our methods to show improved constructions for group testing [Ngo and Du 2000] and generalized hashing [Alon et al. 2003], and an improved inapproximability result for SET-COVER under the assumption P &neq; NP.

On the hardness of approximating the chromatic number

(MATH) We show that the Minimum Vertex Cover problem is NP-hard to approximate to within any factor smaller than

On the complexity of equilibria

10\sqrt{5}-21 \approx 1.36067

On the complexity of approximating k -set packing

, improving on the previously known hardness result for a