License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2022.66
URN: urn:nbn:de:0030-drops-173510
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17351/
Ignatiev, Artur ;
Mihajlin, Ivan ;
Smal, Alexander
Super-Cubic Lower Bound for Generalized Karchmer-Wigderson Games
Abstract
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.
BibTeX - Entry
@InProceedings{ignatiev_et_al:LIPIcs.ISAAC.2022.66,
author = {Ignatiev, Artur and Mihajlin, Ivan and Smal, Alexander},
title = {{Super-Cubic Lower Bound for Generalized Karchmer-Wigderson Games}},
booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
pages = {66:1--66:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-258-7},
ISSN = {1868-8969},
year = {2022},
volume = {248},
editor = {Bae, Sang Won and Park, Heejin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/17351},
URN = {urn:nbn:de:0030-drops-173510},
doi = {10.4230/LIPIcs.ISAAC.2022.66},
annote = {Keywords: communication complexity, circuit complexity, Karchmer-Wigderson games}
}
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Keywords: |
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communication complexity, circuit complexity, Karchmer-Wigderson games |
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Collection: |
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33rd International Symposium on Algorithms and Computation (ISAAC 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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14.12.2022 |
