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 . Previously, the best known tester due to Black, Chakrabarty and Seshadhri  had Ω(n5/6) 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.
|Original language||American English|
|Title of host publication||14th Innovations in Theoretical Computer Science Conference, ITCS 2023|
|Editors||Yael Tauman Kalai|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|Number of pages||24|
|State||Published - 1 Jan 2023|
|Event||14th Innovations in Theoretical Computer Science Conference, ITCS 2023 - Cambridge, United States|
Duration: 10 Jan 2023 → 13 Jan 2023
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||14th Innovations in Theoretical Computer Science Conference, ITCS 2023|
|Period||10/01/23 → 13/01/23|
Bibliographical noteFunding Information:
Funding Mark Braverman: Research supported in part by the NSF Alan T. Waterman Award, Grant No. 1933331, a Packard Fellowship in Science and Engineering, and the Simons Collaboration on Algorithms and Geometry. Subhash Khot: Supported by the NSF Award CCF-1422159, NSF Award CCF-2130816, and the Simons Investigator Award. Guy Kindler: upported by Israel Science Foundation grant no. 2635/19. Dor Minzer: Supported by a Sloan Research Fellowship.
© Mark Braverman, Subhash Khot, Guy Kindler, and Dor Minzer; licensed under Creative Commons License CC-BY 4.0.
- Isoperimetric Inequalities
- Monotonicity Testing
- Property Testing