Improved Monotonicity Testers via Hypercube Embeddings

Authors Mark Braverman, Subhash Khot, Guy Kindler, Dor Minzer

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Author Details

Mark Braverman
  • Department of Computer Science, Princeton University, NJ, USA
Subhash Khot
  • Courant institute of Mathematical Sciences, New York University, NY, USA
Guy Kindler
  • Engineering and Computer Science Department, The Hebrew University, Jerusalem, USA
Dor Minzer
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

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Mark Braverman, Subhash Khot, Guy Kindler, and Dor Minzer. Improved Monotonicity Testers via Hypercube Embeddings. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 25:1-25:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We show improved monotonicity testers for the Boolean hypercube under the p-biased measure, as well as over the hypergrid [m]ⁿ. Our results are: 1) For any p ∈ (0,1), for the p-biased hypercube we show a non-adaptive tester that makes Õ(√n/ε²) queries, accepts monotone functions with probability 1 and rejects functions that are ε-far from monotone with probability at least 2/3. 2) For all m ∈ ℕ, we show an Õ(√nm³/ε²) query monotonicity tester over [m]ⁿ. We also establish corresponding directed isoperimetric inequalities in these domains, analogous to the isoperimetric inequality in [Subhash Khot et al., 2018]. Previously, the best known tester due to Black, Chakrabarty and Seshadhri [Hadley Black et al., 2018] had Ω(n^{5/6}) query complexity. Our results are optimal up to poly-logarithmic factors and the dependency on m. Our proof uses a notion of monotone embeddings of measures into the Boolean hypercube that can be used to reduce the problem of monotonicity testing over an arbitrary product domains to the Boolean cube. The embedding maps a function over a product domain of dimension n into a function over a Boolean cube of a larger dimension n', while preserving its distance from being monotone; an embedding is considered efficient if n' is not much larger than n, and we show how to construct efficient embeddings in the above mentioned settings.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Mathematics of computing → Combinatorics
  • Property Testing
  • Monotonicity Testing
  • Isoperimetric Inequalities


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