Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs.

We introduce a new model for testing graph properties which we call the emph{rejection sampling model}. We show that testing bipartiteness of $n$-nodes graphs using rejection sampling queries requires complexity $widetilde{Omega}(n^2)$. Via reductions from the rejection sampling model, we give three new lower bounds for tolerant testing of Boolean functions of the form $fcolon{0,1}^nto {0,1}$: $bullet$Tolerant $k$-junta testing with emph{non-adaptive} queries requires $widetilde{Omega}(k^2)$ queries. $bullet$Tolerant unateness testing requires $widetilde{Omega}(n)$ queries. $bullet$Tolerant unateness testing with emph{non-adaptive} queries requires $widetilde{Omega}(n^{3/2})$ queries. Given the $widetilde{O}(k^{3/2})$-query non-adaptive junta tester of Blais cite{B08}, we conclude that non-adaptive tolerant junta testing requires more queries than non-tolerant junta testing. In addition, given the $widetilde{O}(n^{3/4})$-query unateness tester of Chen, Waingarten, and Xie cite{CWX17b} and the $widetilde{O}(n)$-query non-adaptive unateness tester of Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri cite{BCPRS17}, we conclude that tolerant unateness testing requires more queries than non-tolerant unateness testing, in both adaptive and non-adaptive settings. These lower bounds provide the first separation between tolerant and non-tolerant testing for a natural property of Boolean functions.