Query-to-Communication Lifting for PNP

Abstract

We prove that the P NP -type query complexity (alternatively, decision list width) of any boolean function f is quadratically related to the P NP -type communication complexity of a lifted version of f . As an application, we show that a certain "product" lower bound method of Impagliazzo and Williams (CCC 2010) fails to capture P NP communication complexity up to polynomial factors, which answers a question of Papakonstantinou, Scheder, and Song (CCC 2014).

Query-to-Communication Lifting for BPP

“…As a minor technicality (for the purpose of deriving Lemma 4 from Lemma 8), we say that a random variable x ∈ [m] J is δ-essentially-dense if for every nonempty I ⊆ J, H ∞ (x I ) ≥ δ • |I| log m − 1 (the difference from Definition 1 is the "−1"); we also define ρ-essentially-structured in the same way as ρ-structured but requiring X free ρ to be only 0.9-essentially-dense instead of 0.9-dense. The following strengthens a lemma from [GKPW17], which implied that G(X, Y ) has full support over the set of all z consistent with ρ.…”

“…Let m = m(n) := n 256 . For every f : {0, 1} n → {0, 1}, decision list rectangle overlay [GKPW17] Sherali-Adams LP extension complexity [CLRS16, KMR17] sum-of-squares SDP extension complexity [LRS15] Table 1: Query-to-communication lifting theorems. The first five are formulated in the language of boolean functions (as in this paper); the last two are formulated in the language of combinatorial optimization.…”

General systems of linear forms: Equidistribution and true complexity

“…Theorem 4.3 (Göös, Kamath, Pitassi, and Watson [GKPW17]). There exists a sequence of 2 n ×2 n Boolean matrices F n satisfying D NP (F n ) ≥ n Ω(1) and log wrect(F n ) −1 ≤ log O(1) (n).…”

“…One might ask whether log wrect(F ) −1 and D NP (F ) are polynomially equivalent. The answer is negative as [GKPW17] constructs an explicit family of Boolean matrices exhibiting a large gap between the two quantities.…”

Untitled

“…For a set X we use the boldface X X X to denote a random variable uniformly distributed over X. 17,22]). For m ≥ n ∆ , every ρ-structured rectangle is ρ-like.…”