Rectangles Are Nonnegative Juntas

Abstract

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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“…It has also been proven in [21] that MA = SBP, solving another open problem posed in the original version of this paper [22]. Thomas …”

“…It has been proven in [21] that SBP( f ) ≤ O(USBP( f ) + log n) for all f (even partial f ) using a new and elementary "Truncation Lemma" which enables rectangle-based techniques to be applied to low nonnegative rank matrices in certain situations. Combining this result with Corollary 1.2 yields an alternative proof of Theorem 1.3 and shows the class equality SBP = USBP.…”

Topology Dependent Bounds For FAQs

“…To the best of our knowledge, min-entropy has only been used very recently in communication complexity [27,28] though it has found numerous applications in pseudorandomness and cryptography for at least two decades [57]. Our work adds to the recently growing body of work that uses min-entropy to prove communication complexity results [17].…”

“…Taking into account that A can be (τ , δ )-bad, we have that A|B (A) = S is ϵ bad (S)-close to an (η, n(1 − η) − log(1/δ ))-block source. 27 . Here, the inequality follows from the fact that any marginal probability values are always at most their original joint probability values.…”

$\mathsf{QMA}$ Lower Bounds for Approximate Counting

“…This suggests that the "Laurent polynomial method" may prove to be useful even for problems involving ordinary polynomials. 1 There even exists an oracle relative to which SBP is not closed under intersection [7], and SBP's closure or non-closure under intersection in the unrelativized world remains an open problem.…”