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Abstract

We construct a simple function that has small unbounded-error communication complexity in the k-party number-on-forehead (NOF) model but every probabilistic protocol that solves it with subexponential advantage over random guessing has cost essentially Ω(√ n/4 k) bits. This separates these classes up to k ≤ δ log n players for any constant δ < 1/4, and gives the largest known separation by an explicit function in this regime of k. Our analysis is elementary and self-contained, inspired by the methods of Goldmann, Håstad, and Razborov (Computational Complexity, 1992). After initial publication of our work as a preprint (ECCC, 2016), Sherstov pointed out to us that an alternative proof of an Ω((n/4 k) 1/7) separation is implicit in his prior work (SICOMP, 2016). Furthermore, based on his prior work (SICOMP, 2013 and SICOMP, 2016), Sherstov gave an alternative proof of our constructive Ω(√ n/4 k) separation and also produced a stronger non-constructive Ω(n/4 k) separation. These results are explained in Sherstov's preprint (ECCC, 2016) and in article 14/22 in Theory of Computing.

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“…Ignoring cosmetic differences, the function in this corollary is the same as the functions used in our original two-party separation [24] and in the proof of Chattopadhyay and Mande [12]. Quantitatively, Corollary 1.2 reproduces the bounds proved by Chattopadhyay and Mande, whereas Theorem 1.1 gives a quadratically stronger nonconstructive result.…”

“…The past few years saw a resurgence of interest in multiparty communication complexity classes, with numerous separations established over the past decade [19,11,14,3,15]. Recently, Chattopadhyay and Mande [12] revisited the unbounded versus weakly unbounded question in the multiparty setting. They generalized the original two-party separation [24] to k 3 parties, exhibiting a k-party communication problem F : ({0, 1} n ) k → {−1, +1} with UPP(F) = O(log n) and PP(F) = Ω( √ n/4 k − log n − k).…”

“…THEORY OF COMPUTING, Volume 14(22), 2018, pp [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. where the maximum is over cylinder intersections χ.…”

The Hardest Halfspace

“…Theorem 1.3 gives essentially the strongest possible explicit separation of the kparty communication complexity classes UPP k and PP k for up to k (0.5 − ) log n parties, where > 0 is an arbitrary constant. The previous best explicit separation [27,80] of these classes was quadratically weaker, with communication complexity Ω( √ n/4 k ) for unbounded error and O(log n) for weakly unbounded error. The communication lower bound in Theorem 1.3 reflects the state of the art in the area, in that the strongest lower bound for any explicit communication problem F : ({0, 1} n ) k → {−1, +1} to date is Ω(n/2 k ) due to Babai et al [12].…”