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STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
On the hardness of approximating N P witnesses
APPROX '00 Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization
List Decoding: Algorithms and Applications
TCS '00 Proceedings of the International Conference IFIP on Theoretical Computer Science, Exploring New Frontiers of Theoretical Informatics
Polynomial Reconstruction Based Cryptography
SAC '01 Revised Papers from the 8th Annual International Workshop on Selected Areas in Cryptography
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Computational Complexity
Private approximation of clustering and vertex cover
TCC'07 Proceedings of the 4th conference on Theory of cryptography
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Structure approximation of most probable explanations in bayesian networks
ECSQARU'13 Proceedings of the 12th European conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty
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We show how to construct proof systems for NP languages where a deterministic polynomial-time verifier can check membership, given any $N^{(2/3)+\epsilon}$ bits of an $N$-bit witness of membership. We also provide a slightly super-polynomial time proof system where the verifier can check membership, given only $N^{(1/2)+\epsilon}$ bits of an $N$-bit witness. These pursuits are motivated by the work of G\'al et. al. [GHLP99]. In addition, we construct proof systems where a deterministic polynomial-time verifier can check membership, given an $N$-bit string that agrees with a legitimate witness on just $(N/2) + N^{(4/5) + \epsilon}$ bits.Our results and framework have applications for two related areas of research in complexity theory: proof systems for $\NP$, and the relative power of Cook reductions and Karp-Levin type reductions. Our proof techniques are based on algebraic coding theory and small sample space constructions.