Communication lower bounds via critical block sensitivity

Göös, Mika; Pitassi, Toniann

doi:10.1145/2591796.2591838

Cited by 35 publications

(39 citation statements)

References 67 publications

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“…We also define MBS (and all other complexity measures introduced later in this section) with respect to real-valued functions over {0, 1} 𝑛 . This differs from block sensitivity, which is usually (though not always [13]) studied in the context of boolean-valued functions. The generalization to real-valued 𝑓 will be immaterial to some of our proofs, permitting us to draw more general conclusions regarding the monomial structure of multilinear polynomials; see Section 5 for more details.…”

Section: Monotone Block Sensitivitymentioning

confidence: 99%

Log-rank and lifting for AND-functions

Knop

Lovett

McGuire

et al. 2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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Let 𝑓 : {0, 1} 𝑛 → {0, 1} be a boolean function, and let 𝑓 ∧ (𝑥, 𝑦) = 𝑓 (𝑥 ∧𝑦) denote the AND-function of 𝑓 , where 𝑥 ∧𝑦 denotes bit-wise AND. We study the deterministic communication complexity of 𝑓 ∧ and show that, up to a log 𝑛 factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of 𝑓 ∧ . This comes within a log 𝑛 factor of establishing the log-rank conjecture for AND-functions with no assumptions on 𝑓 . Our result stands in contrast with previous results on special cases of the logrank conjecture, which needed significant restrictions on 𝑓 such as monotonicity or low F 2 -degree. Our techniques can also be used to prove (within a log 𝑛 factor) a lifting theorem for AND-functions, stating that the deterministic communication complexity of 𝑓 ∧ is polynomially related to the AND-decision tree complexity of 𝑓 .The results rely on a new structural result regarding boolean functions 𝑓 : {0, 1} 𝑛 → {0, 1} with a sparse polynomial representation, which may be of independent interest. We show that if the polynomial computing 𝑓 has few monomials then the set system of the monomials has a small hitting set, of size poly-logarithmic in its sparsity. We also establish extensions of this result to multi-linear polynomials 𝑓 : {0, 1} 𝑛 → R with a larger range. CCS CONCEPTS• Theory of computation → Communication complexity; Oracles and decision trees.

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Section: Monotone Block Sensitivitymentioning

confidence: 99%

Log-rank and lifting for AND-functions

Knop

Lovett

McGuire

et al. 2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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“…We lift the query results of Section 4 to a communication model using the result of [HN12,GP18], Theorem 1. The high-level idea is to use the replication technique (Section 4.1).…”

Section: N-player Hardnessmentioning

confidence: 99%

“…v∈V P [HN12]. and[GP18] have proved that this lifting of the problem to communication is as hard as the query problem, even for randomized communication model.Theorem 1 ([HN12, GP18]). CC(EndOfLine(Pyr(T )) = Θ(QC(EndOfLine(Pyr(T ))) = Θ( √ T ).…”

mentioning

confidence: 99%

Communication complexity of approximate Nash equilibria

Babichenko

Rubinstein

2022

Games and Economic Behavior

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We prove communication complexity lower bounds for (possibly mixed) Nash equilibrium in potential games. In particular, we show that finding a Nash equilibrium requires poly(N) communication in two-player N × N potential games, and 2 poly(n) communication in n-player two-action games. To the best of our knowledge, these are the first results to demonstrate hardness in any model of (possibly mixed) Nash equilibrium in potential games.

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“…In [Göös and Pitassi, 2014] it is shown that there exists a constant degree graph D with N vertices where both the randomized communication complexity of the problem is Θ( √ N ). The proof is done in three steps.…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

The communication complexity of local search

Babichenko

Dobzinski

Nisan

2019

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

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We study the following communication variant of local search. There is some fixed, commonly known graph G. Alice holds f A and Bob holds f B , both are functions that specify a value for each vertex. The goal is to find a local maximum ofOur main result is that finding a local maximum requires polynomial (in the number of vertices) bits of communication. The result holds for the following families of graphs: three dimensional grids, hypercubes, odd graphs, and degree 4 graphs. Moreover, we provide an optimal communication bound of Ω( √ N ) for the hypercube, and for a constant dimensional greed, where N is the number of vertices in the graph.We provide applications of our main result in two domains, exact potential games and combinatorial auctions. First, we show that finding a pure Nash equilibrium in 2-player Naction exact potential games requires polynomial (in N ) communication. We also show that finding a pure Nash equilibrium in n-player 2-action exact potential games requires exponential (in n) communication.The second domain that we consider is combinatorial auctions, in which we prove that finding a local maximum in combinatorial auctions requires exponential (in the number of items) communication even when the valuations are submodular.Each one of the results demonstrates an exponential separation between the non-deterministic communication complexity and the randomized communication complexity of a total search problem. {1, ..., W }, and their goal is to find aDetermining the communication complexity of SumLS on certain families of graphs is easy. For example, a simple reduction from disjointness shows that the communication complexity of SumLS on the clique with n vertices is Ω(n). Our main theorem proves optimal lower bounds for several important families of graphs, all have small degree. The technical challenge is that the nondeterministic communication complexity of the problem on small degree graphs is clearly low: to verify that v * is a local optimum, Alice and Bob need only communicate the values f (u) and g(u) for the small number of v * 's neighbours in the graph (note that the degree of all graphs that we consider is indeed small: log N or even constant). There are only a few results in the communication complexity literature that manage to prove good lower bounds for total problems where verification is easy, most notably for Karchmer-Wigderson games [Karchmer and Wigderson, 1990, Karchmer et al., 1995, Raz and McKenzie, 1997] and for PPAD-like communication problems [Babichenko and Rubinstein, 2016, Göös andRubinstein, 2018].Main Theorem.

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Communication lower bounds via critical block sensitivity

Cited by 35 publications

References 67 publications

Log-rank and lifting for AND-functions

Log-rank and lifting for AND-functions

Communication complexity of approximate Nash equilibria

The communication complexity of local search

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