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--$ ac,J 1Tel-Aviv University, ISRAEL. Permissionto nmhcdigiiol/hmi copies ol’iill w poti ,d’hs motcrizl Ibr personal or claswoom use is grmltd without fee provided Ih:)t the copies ‘are nof m.adt or disirihutcd I“or pI-o~II or ccrmmcrc. ial :Id\’Jm:Igt. Ilw L’OPV right notice. the Iil! c ot’lhc puhiimlion nnd iLs tiate nppcm’, imd uotice is given that copyright is by pcnniwi(m oflhe AChl, 10C. To copyolherwise. 10rcpuhlish. 10 POM m swvws or In redis(rilul! c In lists. requires specilic permission amikrr I’te .STOC ‘ 97 El ]>;lSO. ‘] ’~XSS [ IS.A Copyrighl1997 ACM O-8979 I-XNL61’9705 ,$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-