Today’s post by Clément Canonne.

Following the Boolean monotonicity testing bonanza, here’s an open problem. In short, does adaptivity help for monotonicity testing of Boolean functions?

Problem: Consider the problem of monotonicity testing for Boolean functions on the hypercube. Given oracle access to \(f\colon \{0,1\}^n \to \{0,1\}\), we wish to decide if \(f\) is (i) monotone vs. (ii) \(\epsilon\)-far from monotone (in Hamming distance). For either the one-sided or two-sided version of the problem, what is the exact status of adaptive testers?

State of the art:
– Fischer et al. [FLN+02] showed one-sided non-adaptive testers require \(\sqrt{n}\) queries. This implies an \(\Omega(\log n)\) lower bound for one-sided adaptive testers.
– Chen et al. [CDST15] proved that two-sided non-adaptive testers require (essentially) \(\Omega(\sqrt{n})\) queries. This implies an \(\Omega(\log n)\) lower bound for 2-sided adaptive testers.
– Khot et al. [KMS15] recently gave a one-sided non-adaptive tester making \(\tilde{O}(\sqrt{n}/\epsilon^2)\) queries. The story is essentially complete for non-adaptive testing.

Comments: As of now, it is not clear whether adaptivity can help. Berman et al. [BRY14] showed the benefit of adaptivity for Boolean monotonicity testing over the domain \([n]^2\) (switch the \(2\) and the \(n\) from the hypercube). A gap provably exists between adaptive and non-adaptive testers: \(O(1/\epsilon)\) vs. \(\Omega(\log(1/\epsilon)/\epsilon)\).

References:

[FLN+02] E. Fischer, E. Lehman, I. Newman, S. Raskhodnikova, R. Rubinfeld, and A. Samorodnitsky. Monotonicity testing over general poset domains. Symposium on Theory of Computing, 2002

[BRY14] P. Berman, S. Raskhodnikova, and G. Yaroslavtsev. \(L_p\) testing. Symposium on Theory of Computing, 2014

[CDST15] X. Chen, A. De, R. Servedio, L.-Y. Tang. Boolean function monotonicity testing requires (almost) \(n^{1/2}\) non-adaptive queries. Symposium on Theory of Computing, 2015

[KMS15] S. Khot, D. Minzer, and S. Safra. On monotonicity testing and Boolean Isoperimetric type theorems. ECCC, 2015

Erratum: a previous version of this post stated (incorrectly)  lower bound of \(\Omega(\sqrt{n}/\epsilon^2)\). This has been corrected to \(\Omega(\sqrt{n})\).