Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity

Pitassi, Toniann; Shirley, Morgan; Watson, Thomas

doi:10.1007/s00037-021-00210-5

Cited by 3 publications

(4 citation statements)

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“…Recent work [17] asked whether efficient (i.e. poly(log log 𝑛) on domain [𝑛]) randomized communication can always be simulated by an efficient deterministic protocol with access to an Eqality oracle; we call such protocols equalitybased (see also [12,28,43]). [17] gave a problem that admits an 𝑂 (log log 𝑛) randomized protocol, but any equality-based protocol has cost Ω(log 𝑛).…”

Section: Resultsmentioning

confidence: 99%

“…In this paper we study a new connection between constant-cost randomized 1 communication and a long-standing question about implicit graph representations in structural graph theory [35,52]. Randomized communication is a central topic in communication complexity [17,26,28,43], and the most basic question is, Does a given problem have a randomized protocol with only constant cost? For example, the Eqality problem has constant cost [42], while the Greater-Than problem 2 has non-constant cost.…”

Section: Introductionmentioning

confidence: 99%

“…[16,18,35]), which we call equality-based labeling, is equivalent to constant-cost deterministic communication with an Eqality oracle, which we call equality-based communication. Recent work [12,17,28,43] has compared equality-based communication to randomized communication. We show that equality-based communication cannot compute adjacency in the hypercube.…”

Section: Introductionmentioning

confidence: 99%

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Randomized communication and implicit graph representations

Harms

Wild

Zamaraev

2022

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

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The most basic lower-bound question in randomized communication complexity is: Does a given problem have constant cost, or non-constant cost? We observe that this question has a deep connection to implicit graph representations in structural graph theory. Specifically, constant-cost communication problems correspond to hereditary graph families that admit constant-size adjacency sketches, or equivalently constant-size probabilistic universal graphs (PUGs), and these graph families are a subset of families that admit adjacency labeling schemes of size 𝑂 (log 𝑛), which are the subject of the well-studied implicit graph question (IGQ).We initiate the study of the hereditary graph families that admit constant-size PUGs, with the two (equivalent) goals of (1) giving a structural characterization of randomized constant-cost communication problems, and (2) resolving a probabilistic version of the IGQ. For each family F studied in this paper (including the monogenic bipartite families, product graphs, interval and permutation graphs, families of bounded twin-width, and others), it holds that the subfamilies H ⊆ F are either stable (in a sense relating to model theory), in which case they admit constant-size PUGs (i.e. adjacency sketches), or they are not stable, in which case they do not.The correspondence between communication problems and hereditary graph families allows for a probabilistic method of constructing adjacency labeling schemes. By this method, we show that the induced subgraphs of any Cartesian products 𝐺 𝑑 are positive examples to the IGQ, also giving a bound on the number of unique induced subgraphs of any graph product. We prove that this probabilistic construction cannot be "naïvely derandomized" by using an Eqality oracle, implying that the Eqality oracle cannot simulate the 𝑘-Hamming Distance communication protocol.As a consequence of our results, we obtain constant-size sketches for deciding dist(𝑥, 𝑦) ≤ 𝑘 for vertices 𝑥, 𝑦 in any stable graph family with bounded twin-width, answering an open question about planar graphs from earlier work. This generalizes to constant-size sketches for deciding first-order formulas over the same graphs.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Randomized communication and implicit graph representations

Harms

Wild

Zamaraev

2022

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

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“…Given this exponential gap and lack of progress for general communication problems, many works [3,5,6,[8][9][10][11]13,14,16,[19][20][21]23,26,[29][30][31][32] focused on the communication complexity of functions with some restricted structure. In particular, the study of composed functions was especially successful, and produced the socalled lifting method, which connects query complexity measures of boolean functions with communication complexity measures of their corresponding communication problems.…”

Section: Introductionmentioning

confidence: 99%

Log-rank and lifting for AND-functions

Knop

Lovett

McGuire

et al. 2020

Preprint

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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 log-rank conjecture, which needed significant restrictions on f such as monotonicity or low F2-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.

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