Strongly Exponential Separation between Monotone VP and Monotone VNP

Srikanth Srinivasan

doi:10.1145/3417758

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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Exponential separation and long-time correlation in collinear OCS

Chemical Physics Letters ◽

10.1016/0009-2614(84)87077-3 ◽

1984 ◽

Vol 110 (5) ◽

pp. 491-495 ◽

Cited By ~ 23

Author(s):

Michael J. Davis

Keyword(s):

Time Correlation ◽

Exponential Separation ◽

Long Time

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The randomized communication complexity of revenue maximization

ACM SIGecom Exchanges ◽

10.1145/3505156.3505165 ◽

2021 ◽

Vol 19 (2) ◽

pp. 75-83

Author(s):

Aviad Rubinstein ◽

Junyao Zhao

Keyword(s):

Lower Bounds ◽

Communication Complexity ◽

Social Choice Rule ◽

Valuation Function ◽

Optimal Auctions ◽

Incentive Compatible ◽

First Approximation ◽

Exponential Separation ◽

Open Question

We study the communication complexity of incentive compatible auction-protocols between a monopolist seller and a single buyer with a combinatorial valuation function over n items [Rubinstein and Zhao 2021]. Motivated by the fact that revenue-optimal auctions are randomized [Thanassoulis 2004; Manelli and Vincent 2010; Briest et al. 2010; Pavlov 2011; Hart and Reny 2015] (as well as by an open problem of Babaioff, Gonczarowski, and Nisan [Babaioff et al. 2017]), we focus on the randomized communication complexity of this problem (in contrast to most prior work on deterministic communication). We design simple, incentive compatible, and revenue-optimal auction-protocols whose expected communication complexity is much (in fact infinitely) more efficient than their deterministic counterparts. We also give nearly matching lower bounds on the expected communication complexity of approximately-revenue-optimal auctions. These results follow from a simple characterization of incentive compatible auction-protocols that allows us to prove lower bounds against randomized auction-protocols. In particular, our lower bounds give the first approximation-resistant, exponential separation between communication complexity of incentivizing vs implementing a Bayesian incentive compatible social choice rule, settling an open question of Fadel and Segal [Fadel and Segal 2009].

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