4,085 research outputs found

    Inapproximability of Combinatorial Optimization Problems

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    We survey results on the hardness of approximating combinatorial optimization problems

    On the Power of Many One-Bit Provers

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    We study the class of languages, denoted by \MIP[k, 1-\epsilon, s], which have kk-prover games where each prover just sends a \emph{single} bit, with completeness 1ϵ1-\epsilon and soundness error ss. For the case that k=1k=1 (i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson ({\em Computational Complexity'02}) demonstrate that \SZK exactly characterizes languages having 1-bit proof systems with"non-trivial" soundness (i.e., 1/2<s12ϵ1/2 < s \leq 1-2\epsilon). We demonstrate that for the case that k2k\geq 2, 1-bit kk-prover games exhibit a significantly richer structure: + (Folklore) When s12kϵs \leq \frac{1}{2^k} - \epsilon, \MIP[k, 1-\epsilon, s] = \BPP; + When 12k+ϵs<22kϵ\frac{1}{2^k} + \epsilon \leq s < \frac{2}{2^k}-\epsilon, \MIP[k, 1-\epsilon, s] = \SZK; + When s22k+ϵs \ge \frac{2}{2^k} + \epsilon, \AM \subseteq \MIP[k, 1-\epsilon, s]; + For s0.62k/2ks \le 0.62 k/2^k and sufficiently large kk, \MIP[k, 1-\epsilon, s] \subseteq \EXP; + For s2k/2ks \ge 2k/2^{k}, \MIP[k, 1, 1-\epsilon, s] = \NEXP. As such, 1-bit kk-prover games yield a natural "quantitative" approach to relating complexity classes such as \BPP,\SZK,\AM, \EXP, and \NEXP. We leave open the question of whether a more fine-grained hierarchy (between \AM and \NEXP) can be established for the case when s22k+ϵs \geq \frac{2}{2^k} + \epsilon

    Quantum Weakly Nondeterministic Communication Complexity

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    We study the weakest model of quantum nondeterminism in which a classical proof has to be checked with probability one by a quantum protocol. We show the first separation between classical nondeterministic communication complexity and this model of quantum nondeterministic communication complexity for a total function. This separation is quadratic.Comment: 12 pages. v3: minor correction
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