Reconstructing Weighted Graphs with Minimal Query Complexity

Reconstructing weighted graphs with minimal query complexity

Theoretical Computer Science ◽

10.1016/j.tcs.2010.12.055 ◽

2011 ◽

Vol 412 (19) ◽

pp. 1782-1790 ◽

Cited By ~ 4

Author(s):

Nader H. Bshouty ◽

Hanna Mazzawi

Keyword(s):

Query Complexity ◽

Weighted Graphs

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Constant Rate PCPs for Circuit-SAT with Sublinear Query Complexity

Journal of the ACM ◽

10.1145/2901294 ◽

2016 ◽

Vol 63 (4) ◽

pp. 1-57 ◽

Cited By ~ 4

Author(s):

Eli Ben-Sasson ◽

Yohay Kaplan ◽

Swastik Kopparty ◽

Or Meir ◽

Henning Stichtenoth

Keyword(s):

Query Complexity ◽

Constant Rate

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A Tight Composition Theorem for the Randomized Query Complexity of Partial Functions: Extended Abstract

2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS) ◽

10.1109/focs46700.2020.00031 ◽

2020 ◽

Author(s):

Shalev Ben-David ◽

Eric Blais

Keyword(s):

Query Complexity ◽

Partial Functions ◽

Composition Theorem

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An Exponential Separation Between MA and AM Proofs of Proximity

Computational Complexity ◽

10.1007/s00037-021-00212-3 ◽

2021 ◽

Vol 30 (2) ◽

Author(s):

Tom Gur ◽

Yang P. Liu ◽

Ron D. Rothblum

Keyword(s):

Quantum Information ◽

Lower Bound ◽

Recent Result ◽

Query Complexity ◽

Random String ◽

Natural Property ◽

Public And Private ◽

Alternate Proof ◽

Interactive Proofs ◽

Exponential Separation

AbstractInteractive proofs of proximity allow a sublinear-time verifier to check that a given input is close to the language, using a small amount of communication with a powerful (but untrusted) prover. In this work, we consider two natural minimally interactive variants of such proofs systems, in which the prover only sends a single message, referred to as the proof. The first variant, known as -proofs of Proximity (), is fully non-interactive, meaning that the proof is a function of the input only. The second variant, known as -proofs of Proximity (), allows the proof to additionally depend on the verifier's (entire) random string. The complexity of both s and s is the total number of bits that the verifier observes—namely, the sum of the proof length and query complexity. Our main result is an exponential separation between the power of s and s. Specifically, we exhibit an explicit and natural property $$\Pi$$ Π that admits an with complexity $$O(\log n)$$ O ( log n ) , whereas any for $$\Pi$$ Π has complexity $$\tilde{\Omega}(n^{1/4})$$ Ω ~ ( n 1 / 4 ) , where n denotes the length of the input in bits. Our lower bound also yields an alternate proof, which is more general and arguably much simpler, for a recent result of Fischer et al. (ITCS, 2014). Also, Aaronson (Quantum Information & Computation 2012) has shown a $$\Omega(n^{1/6})$$ Ω ( n 1 / 6 ) lower bound for the same property $$\Pi$$ Π .Lastly, we also consider the notion of oblivious proofs of proximity, in which the verifier's queries are oblivious to the proof. In this setting, we show that s can only be quadratically stronger than s. As an application of this result, we show an exponential separation between the power of public and private coin for oblivious interactive proofs of proximity.

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