Sunflowers and Testing Triangle-Freeness of Functions

Abstract

A function f: Fn2 → {0, 1} is triangle-free if there are no x1, x2, x3 ∈ Fn2 satisfying x1 + x2 + x3 = 0 and f (x1) = f (x2) = f (x3) = 1. In testing triangle-freeness, the goal is to distinguish with high probability triangle-free functions from those that are ε-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 ε (GAFA, 2005), however the best known upper bound is a tower type function of 1/ε. The best known lower bound on the query complexity of the canonical tester is 1/ε13.239 (Fu and Kleinberg, RANDOM, 2014). In this work we introduce a new approach to proving lower bounds on the query com-plexity of triangle-freeness. We relate the problem to combinatorial questions on collections of vectors in ZnD and to sunflower conjectures studied by Alon, Shpilka, and Umans (Comput. Complex., 2013). The relations yield that a refutation of the Weak Sunflower Conjecture over Z4 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 do-main Fnp for every k ≥ 3 and a prime p. In addition, we generalize the lower bound of Fu and Kleinberg to k-cycle-freeness for k ≥ 4 by generalizing the construction of uniquely solvable puzzles due to Coppersmith and Winograd (J. Symbolic Comput., 1990).