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 rejection sampling model. We show that testing bipartiteness of n-nodes graphs using rejection sampling queries requires complexity Ω(^e n²). 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(^e k²) queries. ■ Tolerant unateness testing requires Ω(^e n) queries. ■ Tolerant unateness testing with non-adaptive queries requires Ω(^e n³/²) queries. Given the Õ(k³/²)-query non-adaptive junta tester of Blais [7], we conclude that non-adaptive tolerant junta testing requires more queries than non-tolerant junta testing. In addition, given the Õ(n³/⁴)-query unateness tester of Chen, Waingarten, and Xie [19] and the Õ(n)-query nonadaptive unateness tester of Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri [3], 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.