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).

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Correspondence to Ning Xie.

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Haviv, I., Xie, N. Sunflowers and Testing Triangle-Freeness of Functions. comput. complex. 26, 497–530 (2017). https://doi.org/10.1007/s00037-016-0138-7

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Keywords

  • property testing
  • triangle-freeness
  • sunflowers

Subject classification

  • 68Q17
  • 68Q25
  • 68W20
  • 68W40