Sunflowers and Testing Triangle-Freeness of Functions

Abstract

A function f : F n 2 → {0, 1} is triangle-free if there are no x 1 , x 2 , x 3 ∈ F n 2 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 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/ε 13.239 (Fu and Kleinberg, RANDOM, 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 Z n D 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 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 F n p 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).

The Asymptotic Induced Matching Number of Hypergraphs: Balanced Binary Strings

“…for every k ∈ N 3 using Theorem 3. (The same result was essentially obtained in [17].) In [7] it was moreover shown that Q(Φ (2,2) ) = Q(T (2,2) ) = 2 using Theorem 3.…”

“…A priori, for Φ ⊆ V 1 ו • •×V k we have the upper bound Q(Φ) min i |V i | and therefore it holds that Q(Φ) min i |V i |, since |V ×n i | = |V i | n . Problem 1 has been studied for several families of k-graphs, in several different contexts: the cap set problem [12,33,19,23,24], approaches to fast matrix multiplication [32,4,5,28], arithmetic removal lemmas [21,14], property testing [15,17], quantum information theory [35,36], and the general study of asymptotic properties of tensors [34,7,8]. We finally mention the related result of Ruzsa and Szemerédi which says that the largest subset E ⊆ n 2 such that (E×E×E)∩{({a, b}, {b, c}, {c, a}) : a, b, c ∈ [n]} is a matching, has size n 2−o (1) |E| o(n 2 ) when n goes to infinity [27], see also [2, Equation 2].…”

Universal points in the asymptotic spectrum of tensors

“…Applications of the asymptotic restriction problem include computing the computational complexity of matrix multiplication in algebraic complexity theory [1, 6, 12, 15-17, 31, 33, 39, 40, 53] (see also [7,11,30,32]), deciding the feasibility of an asymptotic transformation between pure quantum states via stochastic local operations and classical communication (slocc) in quantum information theory [3,19,27,50], bounding the size of combinatorial structures like cap sets and tri-colored sum-free sets in additive combinatorics [2,8,18,20,21,29,[46][47][48], and bounding the query complexity of certain properties in algebraic property testing [4,5,24,26,28,38]. We will elaborate on these connections later (Section 3).…”

A lower bound for the k‐multicolored sum‐free problem in Zmn

“…It would be interesting to close this gap and determine the optimal exponent for the arithmetic k-cycle removal lemma in F n p . Arithmetic removal lemmas were introduced by Green in 2005 [20], and since then the problem of improving the bounds in arithmetic removal lemmas has been widely studied [6,7,14,15,18,22]. This is in part due to the close connection to property testing.…”