2026-07-21T14:16:52Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17351 2024-03-06T10:59:32Z ddc:004 open_access

Super-Cubic Lower Bound for Generalized Karchmer-Wigderson Games Ignatiev, Artur Mihajlin, Ivan Smal, Alexander communication complexity circuit complexity Karchmer-Wigderson games In this paper, we prove a super-cubic lower bound on the size of a communication protocol for generalized Karchmer-Wigderson game for an explicit function f: {0,1}ⁿ → {0,1}^{log n}. Lower bounds for original Karchmer-Wigderson games correspond to De Morgan formula lower bounds, thus the best known size lower bound is cubic. The generalized Karchmer-Wigderson games are similar to the original ones, so we hope that our approach can provide an insight for proving better lower bounds on the original Karchmer-Wigderson games, and hence for proving new lower bounds on De Morgan formula size. To achieve super-cubic lower bound we adapt several techniques used in formula complexity to communication protocols, prove communication complexity lower bound for a composition of several functions with a multiplexer relation, and use a technique from [Ivan Mihajlin and Alexander Smal, 2021] to extract the "hardest" function from it. As a result, in this setting we are able to show that there is a relatively small set of functions such that at least one of them does not have a small protocol. The resulting lower bound of Ω̃(n^3.156) is significantly better than the bound obtained from the counting argument. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Artur Ignatiev and Ivan Mihajlin and Alexander Smal 2022 Is Part Of LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2022.66 urn:nbn:de:0030-drops-173510 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.66 eng https://creativecommons.org/licenses/by/4.0/legalcode