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A Directed Isoperimetric Theorem for Boolean Functions over Hypergrids with Applications to Monotonicity Testing

  • Author(s): Black, Hadley
  • Advisor(s): Seshadhri, C.
  • et al.
Abstract

We study monotonicity testing of Boolean functions over the hypergrid [n]^d and design a non-adaptive tester with 1-sided error whose query complexity is O(d^{5/6}) poly(log n,1/epsilon). Previous to our work, the best known testers had query complexity linear in d but independent of n. We improve upon these testers as long as n = 2^{d^{o(1)}}.

To obtain our results, we work with what we call the augmented hypergrid, which adds extra edges to the hypergrid. Our main technical contribution is a Margulis-style isoperimetric result for the augmented hypergrid, and our tester, like previous testers for the hypercube domain, performs directed random walks on this structure.

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