TY - GEN
T1 - Lower bounds for tolerant junta and unateness testing via rejection sampling of graphs
AU - Levi, Amit
AU - Waingarten, Erik
N1 - Publisher Copyright: © Amit Levi and Erik Waingarten.
PY - 2019/1/1
Y1 - 2019/1/1
N2 - 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.
AB - 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.
KW - Boolean functions
KW - Juntas
KW - Property Testing
KW - Tolerant Testing
UR - http://www.scopus.com/inward/record.url?scp=85069509330&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ITCS.2019.52
DO - 10.4230/LIPIcs.ITCS.2019.52
M3 - Conference contribution
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 10th Innovations in Theoretical Computer Science, ITCS 2019
A2 - Blum, Avrim
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 10th Innovations in Theoretical Computer Science, ITCS 2019
Y2 - 10 January 2019 through 12 January 2019
ER -