2026-07-22T06:32:25Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:15524 2024-03-06T10:55:30Z ddc:004 open_access

Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture Chattopadhyay, Arkadev Garg, Ankit Sherif, Suhail Approximate Rank Randomized Parity Decision Trees Randomized Communication Complexity XOR functions Subspace Designs We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on n input bits, each of which has approximate Fourier sparsity at most O(n³) and randomized parity decision tree complexity Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif [Chattopadhyay et al., 2020] both qualitatively (in terms of designing a large number of examples) and quantitatively (shrinking the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Arkadev Chattopadhyay and Ankit Garg and Suhail Sherif 2021 Is Part Of LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.FSTTCS.2021.13 urn:nbn:de:0030-drops-155245 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.13 eng https://creativecommons.org/licenses/by/4.0/legalcode