The problem of testing monotonicity for Boolean functions on the hypergrid, $f:[n]^d \to \{0,1\}$ is a classic topic in property testing. When $n=2$, the domain is the hypercube. For the hypercube case, a breakthrough result of Khot-Minzer-Safra (FOCS 2015) gave a non-adaptive, one-sided tester making $\otilde(\eps^{-2}\sqrt{d})$ queries. Up to polylog $d$ and $\eps$ factors, this bound matches the $\widetilde{\Omega}(\sqrt{d})$-query non-adaptive lower bound (Chen-De-Servedio-Tan (STOC 2015), Chen-Waingarten-Xie (STOC 2017)). For any $n > 2$, the optimal non-adaptive complexity was unknown. A previous result of the authors achieves a $\otilde(d^{5/6})$-query upper bound (SODA 2020), quite far from the $\sqrt{d}$ bound for the hypercube. In this paper, we resolve the non-adaptive complexity of monotonicity testing for all constant $n$, up to $\poly(\eps^{-1}\log d)$ factors. Specifically, we give a non-adaptive, one-sided monotonicity tester making $\otilde(\eps^{-2}n\sqrt{d})$ queries. From a technical standpoint, we prove new directed isoperimetric theorems over the hypergrid $[n]^d$. These results generalize the celebrated directed Talagrand inequalities that were only known for the hypercube.
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This content will become publicly available on January 30, 2025
Property testing with online adversaries
The online manipulation-resilient testing model, proposed by Kalemaj, Raskhodnikova and Varma
(ITCS 2022 and Theory of Computing 2023), studies property testing in situations where access
to the input degrades continuously and adversarially. Specifically, after each query made by the
tester is answered, the adversary can intervene and either erase or corrupt t data points. In this
work, we investigate a more nuanced version of the online model in order to overcome old and new
impossibility results for the original model. We start by presenting an optimal tester for linearity and
a lower bound for low-degree testing of Boolean functions in the original model. We overcome the
lower bound by allowing batch queries, where the tester gets a group of queries answered between
manipulations of the data. Our batch size is small enough so that function values for a single batch
on their own give no information about whether the function is of low degree. Finally, to overcome
the impossibility results of Kalemaj et al. for sortedness and the Lipschitz property of sequences, we
extend the model to include t < 1, i.e., adversaries that make less than one erasure per query. For
sortedness, we characterize the rate of erasures for which online testing can be performed, exhibiting
a sharp transition from optimal query complexity to impossibility of testability (with any number of
queries). Our online tester works for a general class of local properties of sequences. One feature of
our results is that we get new (and in some cases, simpler) optimal algorithms for several properties
in the standard property testing model.
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- Award ID(s):
- 2022448
- PAR ID:
- 10524532
- Publisher / Repository:
- Proceedings of the 14th Innovations in Theoretical Computer Science Conference (ITCS)
- Date Published:
- Format(s):
- Medium: X
- Location:
- Berkeley, California
- Sponsoring Org:
- National Science Foundation
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