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 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.

Rectangles Are Nonnegative Juntas

“…In fact, the separation of [8] already witnesses PostBPP cc UPostBPP cc where the latter is the unrestricted version of the former. By contrast, we prove that the distinction is immaterial for WAPP and SBP, even for partial functions: the unrestricted models UWAPP cc and USBP cc (see [16] for definitions) are essentially no more powerful than their restricted counterparts. Consequently, the Simulation Theorem can be applied to analyze these unrestricted models, too-but the equivalences are also interesting in their own right.…”

“…Our discussion in this extended abstract is somewhat informal; see the full version [16] for precise definitions.…”

“…(There is one subtlety concerning the two-sided model PostBPP; see the full version [16] for more details. ) We also mention that the result for the simplest model C = NP does not require the full power of the Junta Theorem: it is possible to prove it using only a proper subset of the ideas that we present for the other randomized models.…”

Untitled

“…The simulation theorem asserts that the optimal protocol for f • g n is to simulate the decision tree that computes f if g is a hard function. Simulation theorems have been established in various cases, when g is bitwise AND or OR [15], Inner-Product [3], Index Function [13,4]. Our work gives a new simulation theorem when g is an XOR function.…”

“…After almost a decade of efforts, the conjecture has been established for several classes of XOR-function, such as symmetric functions [20], monotone functions and linear threshold functions [12], constant F 2 -degree functions [17]. A different line of work close to ours is the simulation theorem in [13,20,15,3,4]. They study the relationship between the (regular) decision tree complexity of function f and the communication complexity of f • g n where g is a 2-argument function of small size.…”