Eecient Probabilistically Checkable Proofs and Applications to Approximation

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 (logn) implies NP DTIME(n ); approximating the maximum of a quartic program within any constant is NP-hard; approximating maximum clique within n implies NP BPP; approximating chromatic number within n implies NP BPP; and approximating MAX3SAT within 113=112 is NP-complete. Department of Computer Science & Engineering, Mail Code 0114, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093. E-mail: [email protected]. y MIT Laboratory for Computer Science, 545 Technology Square, Cambridge, MA 02139, USA. e-mail: shafi@theory. lcs.mit.edu. Partially supported by NSF FAW grant No. 9023312-CCR, DARPA grant No. N00014-92-J-1799, and grant No. 89-00312 from the United States Israel Binational Science Foundation (BSF), Jerusalem, Israel. z AT&T Bell Laboratories, Room 2C324, 600 Mountain Avenue, P. O. Box 636, Murray Hill, NJ 07974-0636, USA. email: [email protected]. x MIT Laboratory for Computer Science, 545 Technology Square, Cambridge, MA 02139, USA. e-mail: [email protected]. mit.edu. Supported by a NSF Graduate Fellowship and by NSF grant 92-12184, AFOSR 89-0271, and DARPA N00014-92-J-1799.