Sunflowers and Testing Triangle-Freeness of Functions

Abstract

A function ${f : {\mathbb F}_{2}^{n} \rightarrow {\{0,1\}}}$ is triangle-free if there are no ${x_{1},x_{2},x_{3} \in {\mathbb F}_{2}^{n}}$ satisfying ${x_{1} + x_{2} + x_{3} = 0}$ and ${f(x_{1}) = f(x_{2}) = f(x_{3}) = 1}$. In testing triangle-freeness, the goal is to distinguish with high probability triangle-free functions from those that are ${\varepsilon}$-far from being triangle-free. It was shown by Green that the query complexity of the canonical tester for the problem is upper bounded by a function that depends only on ${\varepsilon}$ (Green 2005); however, the best-known upper bound is a tower-type function of ${1/\varepsilon}$. The best known lower bound on the query complexity of the canonical tester is ${1/\varepsilon^{13.239}}$ (Fu & Kleinberg 2014).

In this work we introduce a new approach to proving lower bounds on the query complexity of triangle-freeness. We relate the problem to combinatorial questions on collections of vectors in ${{\mathbb Z}_D^n}$ and to sunflower conjectures studied by Alon, Shpilka & Umans (2013). The relations yield that a refutation of the Weak Sunflower Conjecture over ${{\mathbb Z}_{4}}$ implies a super-polynomial lower bound on the query complexity of the canonical tester for triangle-freeness. Our results are extended to testing k-cycle-freeness of functions with domain ${{\mathbb F}_p^n}$ for every ${k \ge 3}$ and a prime p. In addition, we generalize the lower bound of Fu and Kleinberg to k-cycle-freeness for ${k \geq 4}$ by generalizing the construction of uniquely solvable puzzles due to Coppersmith & Winograd (1990).