The unbounded-error communication complexity of symmetric functions

Sherstov, Alexander A.

doi:10.1007/s00493-011-2580-0

Cited by 20 publications

(15 citation statements)

References 40 publications

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“…See also the survey [45]. Both methods have since been studied in the two-party setting [35,43,48] and also the multiparty setting [33,3,11,50,51,38].…”

Section: Main Structural Result: Junta Theoremmentioning

confidence: 98%

Rectangles Are Nonnegative Juntas

Göös

Lovett

Meka

et al. 2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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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 the strongest yet 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 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.

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“…See also the survey [45]. Both methods have since been studied in the two-party setting [35,43,48] and also the multiparty setting [33,3,11,50,51,38].…”

Section: Main Structural Result: Junta Theoremmentioning

confidence: 98%

Rectangles Are Nonnegative Juntas

Göös

Lovett

Meka

et al. 2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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“…See also the survey [She08]. Both methods have since been studied in the two-party setting [LZ10,RS10,She11b] and also the multiparty setting [LS09b,AC08,Cha08,She12b,She14,RY15].…”

Section: Main Structural Result: Junta Theoremmentioning

confidence: 99%

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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“…We note that the case ℓ = 2 of Theorem 2.17 is vastly easier to prove than the general statement; this two-party result can be found in [44, Theorem 7.3 and equation (7.3)]. For our sign-rank lower bounds, we use the following theorem implicit in [45]. .…”

Section: Communication Complexitymentioning

confidence: 99%

“…The purpose of this appendix is to prove Theorem 2.18, implicit in [45,39]. We closely follow the treatment in those earlier papers.…”

Section: Consider the Probability Distributionsmentioning

confidence: 99%

Near-optimal lower bounds on the threshold degree and sign-rank of AC ⁰

Sherstov

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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The threshold degree of a Boolean function f : {0, 1} n → {0, 1} is the minimum degree of a real polynomial p that represents f in sign: sgn p(x) = (−1) f (x) . A related notion is sign-rank, defined for a Boolean matrix F = [F ij ] as the minimum rank of a real matrix M with sgn M ij = (−1) F ij . Determining the maximum threshold degree and sign-rank achievable by constant-depth circuits (AC 0 ) is a well-known and extensively studied open problem, with complexity-theoretic and algorithmic applications.We give an essentially optimal solution to this problem. For any ǫ > 0, we construct an AC 0 circuit in n variables that has threshold degree Ω(n 1−ǫ ) and sign-rank exp(Ω(n 1−ǫ )), improving on the previous best lower bounds of Ω( √ n) and exp(Ω( √ n)), respectively. Our results subsume all previous lower bounds on the threshold degree and sign-rank of AC 0 circuits of any given depth, with a strict improvement starting at depth 4. As a corollary, we also obtain near-optimal bounds on the discrepancy, threshold weight, and threshold density of AC 0 , strictly subsuming previous work on these quantities. Our work gives some of the strongest lower bounds to date on the communication complexity of AC 0 .

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The unbounded-error communication complexity of symmetric functions

Cited by 20 publications

References 40 publications

Rectangles Are Nonnegative Juntas

Rectangles Are Nonnegative Juntas

Rectangles Are Nonnegative Juntas

Near-optimal lower bounds on the threshold degree and sign-rank of AC ⁰

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The unbounded-error communication complexity of symmetric functions

Cited by 20 publications

References 40 publications

Rectangles Are Nonnegative Juntas

Rectangles Are Nonnegative Juntas

Rectangles Are Nonnegative Juntas

Near-optimal lower bounds on the threshold degree and sign-rank of AC 0

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Near-optimal lower bounds on the threshold degree and sign-rank of AC ⁰