Two prover protocols: low error at affordable rates

We introduce the miss-match form for two-prover one-round proof systems.Any two-prover one-round proof system can be easily modified so as to be in miss-match form.Proof systems in miss-match form have the "projection" property that is important for deriving hardness of approximation results for NP-hard combinatorial optimization problems. Our main result is an upper bound on the number of parallel repetitions that suffice in order to reduce the error of miss-match proof systems from p to � .This upper bound depends only on p and on � (polynomial in 1/(1 − p) and in 1/� ).Based on previous work, it follows that for any �> 0, NP has two-prover one-round proof systems with logarithmic-sized questions, constant-sized answers, and error at most � . As part of our proof we prove upper bounds on the influence of random variables on multivariate functions, which may be of independent interest.

[1]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[2]  Silvio Micali,et al.  The knowledge complexity of interactive proof-systems , 1985, STOC '85.

[3]  Avi Wigderson,et al.  Multi-prover interactive proofs: how to remove intractability assumptions , 2019, STOC '88.

[4]  Nathan Linial,et al.  The influence of variables on Boolean functions , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[5]  Avi Wigderson,et al.  Efficient Identification Schemes Using Two Prover Interactive Proofs , 1989, CRYPTO.

[6]  Richard J. Lipton,et al.  Playing Games of Incomplete Information , 1990, Symposium on Theoretical Aspects of Computer Science.

[7]  Nondeterministic exponential time has two-prover interactive protocols , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[8]  Richard J. Lipton,et al.  On bounded round multiprover interactive proof systems , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[9]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[10]  Richard J. Lipton,et al.  PSPACE is provable by two provers in one round , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[11]  Adi Shamir,et al.  Fully parallelized multi prover protocols for NEXP-time , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[12]  Leonid A. Levin,et al.  Checking computations in polylogarithmic time , 1991, STOC '91.

[13]  F. Leighton,et al.  Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes , 1991 .

[14]  Uriel Feige.  On the success probability of the two provers in one-round proof systems , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[15]  Moni Naor,et al.  Low Communication 2-Prover Zero-Knowledge Proofs for NP , 1992, CRYPTO.

[16]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[17]  L. Roniger,et al.  From pioneer to freier: The changing models of generalized exchange in Israel , 1992, European Journal of Sociology.

[18]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[19]  László Lovász,et al.  Two-prover one-round proof systems: their power and their problems (extended abstract) , 1992, STOC '92.

[20]  Piotr Berman,et al.  On the Complexity of Approximating the Independent Set Problem , 1989, Inf. Comput..

[21]  Carsten Lund,et al.  Efficient probabilistically checkable proofs and applications to approximations , 1993, STOC.

[22]  Nimrod Megiddo,et al.  Constructing small sample spaces satisfying given constraints , 1993, SIAM J. Discret. Math..

[23]  Richard J. Lipton,et al.  PSPACE is provable by two provers in one round , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[24]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1994, JACM.

[25]  Oleg Verbitsky,et al.  Towards the parallel repetition conjecture , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[26]  Mihir Bellare,et al.  Improved non-approximability results , 1994, STOC '94.

[27]  Carsten Lund,et al.  Efficient probabilistic checkable proofs and applications to approximation , 1994, STOC '94.

[28]  Lance Fortnow,et al.  On the Power of Multi-Prover Interactive Protocols , 1994, Theor. Comput. Sci..

[29]  U. Feige.  Error reduction by parallel repetition-the state of the art , 1995 .

[30]  Adi Shamir,et al.  A one-round, two-prover, zero-knowledge protocol for NP , 1995, Comb..

[31]  J. Kilian,et al.  On the complexity of statistical reasoning , 1995, Proceedings Third Israel Symposium on the Theory of Computing and Systems.

[32]  Ran Raz,et al.  A parallel repetition theorem , 1995, STOC '95.

[33]  Mihir Bellare,et al.  The complexity of approximating a nonlinear program , 1995, Math. Program..