Efficient probabilistically checkable proofs and applications to approximations

Abstract

Efficient Probabilistically Checkable Proofs and Applications to Approximation M. BELLARE* S. GOLDWASSERt C. LUNDi A. RUSSELL$ We construct multi-prover proof systems for NP which use only a constant number of provers to simultaneously achieve low error, low randomness and low answer size. As a consequence, we obtain asymptotic improvements to approximation hardness results for a wide range of optimization problems including minimum set cover, dominating set, maximum clique, chromatic number, and quartic programming; and constant factor improvements on the hardness results for MAXSNP problems. In particular, we show that approximating minimum set cover within any constant is NP-complete; approximating minimum set cover within c log n, for c < 1/8, implies NP C DTIME(nlOglOgn); approximat— ing the maximum of a quartic program within any constant is NP-complete; approximating maximum clique or chromatic number within nl/29 implies NP ~ BPP; and approximating MAX-3 SAT within 113/112 is NPcomplete. * High Performance Computing and Communications, IBM T.J. Watson Research Center, PO Box 704, Yorktown Heights, NY 10598, USA. e-mail: mihirf.Qwatson. ibm. corn. t MIT Laboratory for Computer Science, 545 Technology Square, Cambridge, MA 02139, USA. e-mail: shaf i@theory. lcs. init. edu. Partially supported by NSF FAW grant No. 9023312-CCR, DARPA g-rant No. NOO014-92-J-1799, and grant No. 89-00312 from the United States Israel Binationsl Science Foundation (BSF), Jerusalem, Israel. $ AT&T Bell Laboratories, Room 2C324, 600 Momtain Avenue, P. O. Box 636, Murray Hill, NJ 07974-0636, USA. email: lund@resesrch. att. corn. $ MIT Laboratory for Computer Science, 545 Technology Square, Cambridge, MA 02139, USA. e-mail: acrtttheory. lcs. mit . edn. Supported by a NSF Graduate Fellowship and by NSF grant 92-12184, AFOSR 89-0271, and DARPA NOO014-92-J-1799. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 25th ACM STOC ‘93-51931CA,USA