Uniform Generation of NP-Witnesses Using an NP-Oracle

A uniform generation procedure for NP is an algorithm that, given any input in a fixed NP-language, outputs a uniformly distributed NP-witness for membership of the input in the language. We present a uniform generation procedure for NP that runs in probabilistic polynomial time with an NP-oracle. This improves upon results of M. Jerrum et al. (1986, Theoret. Comput. Sci.43, 169?188), which either require a ?P2 oracle or obtain only almost uniform generation. Our procedure utilizes ideas originating in the works of M. Sipser, and L. Stockmeyer (respectively, 1983, in Proceedings of the 15th Annual Symposium on the Theory of Computing, ACM, New York), and Jerrum et al. (1986).

[1]  Rafail Ostrovsky,et al.  Computational Complexity and Knowledge Complexity , 1994, Electron. Colloquium Comput. Complex..

[2]  Leslie G. Valiant,et al.  NP is as easy as detecting unique solutions , 1985, STOC '85.

[3]  Richard M. Karp,et al.  Monte-Carlo algorithms for enumeration and reliability problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[4]  Oded Goldreich,et al.  Quantifying knowledge complexity , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[5]  Mihir Bellare,et al.  Making zero-knowledge provers efficient , 1992, STOC '92.

[6]  Rafail Ostrovsky,et al.  The (true) complexity of statistical zero knowledge , 1990, STOC '90.

[7]  Moni Naor,et al.  The Probabilistic Method Yields Deterministic Parallel Algorithms , 1994, J. Comput. Syst. Sci..

[8]  Johan Håstad,et al.  Statistical Zero-Knowledge Languages can be Recognized in Two Rounds , 1991, J. Comput. Syst. Sci..

[9]  Leslie G. Valiant,et al.  Random Generation of Combinatorial Structures from a Uniform Distribution , 1986, Theor. Comput. Sci..

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

[11]  Michael Sipser,et al.  A complexity theoretic approach to randomness , 1983, STOC.

[12]  Mihir Bellare,et al.  On the role of shared randomness in two prover proof systems , 1995, Proceedings Third Israel Symposium on the Theory of Computing and Systems.

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

[14]  Oded Goldreich,et al.  On the Complexity of Interactive Proofs with Bounded Communication , 1998, Inf. Process. Lett..

[15]  Aravind Srinivasan,et al.  Chernoff-Hoeffding bounds for applications with limited independence , 1995, SODA '93.

[16]  Larry J. Stockmeyer,et al.  The complexity of approximate counting , 1983, STOC.

[17]  Shafi Goldwasser,et al.  Private coins versus public coins in interactive proof systems , 1986, STOC '86.

[18]  Bonnie Berger,et al.  Simulating (log/sup c/n)-wise independence in NC , 1989, 30th Annual Symposium on Foundations of Computer Science.

[19]  Gábor Tardos,et al.  On the Knowledge Complexity of NP , 1996, IEEE Annual Symposium on Foundations of Computer Science.

[20]  Mihir Bellare,et al.  Randomness-efficient oblivious sampling , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[21]  Lance Fortnow,et al.  The Complexity of Perfect Zero-Knowledge , 1987, Proceeding Structure in Complexity Theory.

[22]  Victor Shoup,et al.  New algorithms for finding irreducible polynomials over finite fields , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[23]  Carsten Lund,et al.  Non-deterministic exponential time has two-prover interactive protocols , 2005, computational complexity.