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Göös, Mika; Watson, Thomas

doi:10.4086/toc.2016.v012a009

Cited by 4 publications

(10 citation statements)

References 44 publications

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“…Namely, Klauck [Kla03] asked (using the language of uniform threshold covers) whether the known inclusion MA cc ⊆ SBP cc is strict. (This was also re-asked in [GW16].) Put diffferently, is corruption a complete lower bound method for MA cc up to polynomial factors?…”

Section: Applicationsmentioning

confidence: 96%

“…An SBP computation is a randomized computation such that 1-inputs are accepted with probability in [α, 1] and 0-inputs are accepted with probability in [0, α/2] where α = α(n) > 0 is arbitrary. The communication analog SBP cc is equivalent to the (one-sided) corruption bound originally defined in [Yao83] (see [GW16]).…”

Section: Communication Versus Query: Simulation Theoremmentioning

confidence: 99%

“…One can analogously ask whether the unrestricted-restricted distinction is relevant for the models considered in this work. (The question was raised and left unresolved for SBP in [GW16].) In fact, the separation of [BVdW07]…”

Section: Applicationsmentioning

confidence: 99%

“…It is straightforward to see that the relative computational power ("semantic relationships") of the models is as follows (recall Figure 1): for all F and all constants 0 < < 1/2, we have 2WAPP cc (F ) ≥ WAPP cc (F ) ≥ Ω(SBP cc (F )) ≥ Ω(PostBPP cc (F )) and NP cc (F ) ≥ SBP cc (F ). Furthermore, exponential separations are known for all these relationships: unique-set-intersection is easy for WAPP cc 0 but hard for 2WAPP cc (indeed, for coSBP cc [Raz92, GW16]); set-intersection is easy for SBP cc (indeed, for NP cc ) but hard for WAPP cc [Kla10]; set-disjointness is easy for PostBPP cc (indeed, for coNP cc ) but hard for SBP cc [Raz92,GW16]; equality is easy for SBP cc (indeed, for coRP cc ) but hard for NP cc . Moreover, WAPP cc is a one-sided version of 2WAPP cc in the sense that 2WAPP cc (F ) ≤ O(WAPP cc /2 (F ) + coWAPP cc /2 (F )) (so the classes would satisfy 2WAPP cc = WAPP cc ∩ coWAPP cc if we ignore the precise value of the constant ).…”

Section: Two-sidedmentioning

confidence: 99%

“…Such models can be used to capture all known rectangle-based lower bound techniques used in communication complexity. This includes widely studied measures such as corruption [Yao83,BFS86,Raz92,Kla03,BPSW06,She12a,GW16], smooth rectangle bound [JK10,Kla10,CKW12,JY12,HJ13,KMSY14], relaxed partition bound [KLL + 15], and extended discrepancy [Kla03,GL14]; see [JK10] for an extensive catalog. The Simulation Theorem applies to all these measures: it reduces the task of understanding a specific communication complexity measure of f • g n to the task of understanding a corresponding query complexity measure of f , which is typically a far easier task.…”

Section: Introductionmentioning

confidence: 99%

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Rectangles Are Nonnegative Juntas

Göös¹,

Lovett²,

Meka³

et al. 2016

SIAM J. Comput.

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We develop a new method to prove communication lower bounds for composed functions of the form f • g n where f is any boolean function on n inputs and g is a sufficiently "hard" two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of f • g n can be simulated by a nonnegative combination of juntas. This is a new formalization for the intuition that each low-communication randomized protocol can only "query" few inputs of f as encoded by the gadget g. Consequently, we characterize the communication complexity of f • g n in all known one-sided (i.e., not closed under complement) zero-communication models by a corresponding query complexity measure of f. These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work: We show that SBP cc (a class characterized by corruption) is not closed under intersection. An immediate corollary is that MA cc = SBP cc. These results answer questions of Klauck (CCC 2003) and Böhler et al. (JCSS 2006). We also show that approximate nonnegative rank of partial boolean matrices does not admit efficient error reduction. This answers a question of Kol et al. (ICALP 2014) for partial matrices. In subsequent work, our structure theorem has been applied to resolve the communication complexity of the Clique vs. Independent Set problem.

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Section: Applicationsmentioning

confidence: 96%

Section: Communication Versus Query: Simulation Theoremmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

Section: Two-sidedmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rectangles Are Nonnegative Juntas

Göös¹,

Lovett²,

Meka³

et al. 2016

SIAM J. Comput.

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A Lower Bound for Sampling Disjoint Sets

Göös

Watson

2020

ACM Trans. Comput. Theory

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Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x ⊆ [ n ] and Bob ends up with a set y ⊆ [ n ], such that ( x , y ) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant β < 1, this requires Ω ( n ) communication even to get within statistical distance 1− β n of the target distribution. Previously, Ambainis, Schulman, Ta-Shma, Vazirani, and Wigderson (FOCS 1998) proved that Ω (√ n ) communication is required to get within some constant statistical distance ɛ > 0 of the uniform distribution over all pairs of disjoint sets of size √ n .

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Quantum states cannot be transmitted efficiently classically

Montanaro¹

2019

Quantum

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We show that any classical two-way communication protocol with shared randomness that can approximately simulate the result of applying an arbitrary measurement (held by one party) to a quantum state of n qubits (held by another), up to constant accuracy, must transmit at least Ω(2 n ) bits. This lower bound is optimal and matches the complexity of a simple protocol based on discretisation using an -net. The proof is based on a lower bound on the classical communication complexity of a distributed variant of the Fourier sampling problem. We obtain two optimal quantum-classical separations as easy corollaries. First, a sampling problem which can be solved with one quantum query to the input, but which requires Ω(N ) classical queries for an input of size N . Second, a nonlocal task which can be solved using n Bell pairs, but for which any approximate classical solution must communicate Ω(2 n ) bits.1 Note that the complexity achieved by this approach is better than would be obtained by simply writing down |ψ in the computational basis, and truncating each amplitude after some number of digits. To achieve sufficient precision with this approach requires specifying each amplitude up to precision O( / √ 2 n ), giving an overall communication complexity of O(2 n log(2 n / )).

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Cited by 4 publications

References 44 publications

Rectangles Are Nonnegative Juntas

Rectangles Are Nonnegative Juntas

A Lower Bound for Sampling Disjoint Sets

Quantum states cannot be transmitted efficiently classically

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