Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs
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We introduce a new model for testing graph properties which we call the rejection sampling model. We show that testing bipartiteness of n-nodes graphs using rejection sampling queries requires complexity Ω(n^2). Via reductions from the rejection sampling model, we give three new lower bounds for tolerant testing of Boolean functions of the form f{0,1}^n→{0,1}: ∙Tolerant k-junta testing with non-adaptive queries requires Ω(k^2) queries. ∙Tolerant unateness testing requires Ω(n) queries. ∙Tolerant unateness testing with non-adaptive queries requires Ω(n^3/2) queries. Given the O(k^3/2)-query non-adaptive junta tester of Blais B08, we conclude that non-adaptive tolerant junta testing requires more queries than non-tolerant junta testing. In addition, given the O(n^3/4)-query unateness tester of Chen, Waingarten, and Xie CWX17b and the O(n)-query non-adaptive unateness tester of Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri 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.
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