# Open problem for February 2015

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 $$\Omega(\sqrt{n})$$.