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.CCC.2021.38
URN: urn:nbn:de:0030-drops-143121
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14312/
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Mihajlin, Ivan ; Smal, Alexander

Toward Better Depth Lower Bounds: The XOR-KRW Conjecture

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LIPIcs-CCC-2021-38.pdf (0.9 MB)

Abstract

In this paper, we propose a new conjecture, the XOR-KRW conjecture, which is a relaxation of the Karchmer-Raz-Wigderson conjecture [Mauricio Karchmer et al., 1995]. This relaxation is still strong enough to imply ? ̸ ⊆ NC¹ if proven. We also present a weaker version of this conjecture that might be used for breaking n³ lower bound for De Morgan formulas. Our study of this conjecture allows us to partially answer an open question stated in [Dmitry Gavinsky et al., 2017] regarding the composition of the universal relation with a function. To be more precise, we prove that there exists a function g such that the composition of the universal relation with g is significantly harder than just a universal relation. The fact that we can only prove the existence of g is an inherent feature of our approach.
The paper’s main technical contribution is a new approach to lower bounds for multiplexer-type relations based on the non-deterministic hardness of non-equality and a new method of converting lower bounds for multiplexer-type relations into lower bounds against some function. In order to do this, we develop techniques to lower bound communication complexity in half-duplex and partially half-duplex communication models.

BibTeX - Entry

@InProceedings{mihajlin_et_al:LIPIcs.CCC.2021.38,
  author =	{Mihajlin, Ivan and Smal, Alexander},
  title =	{{Toward Better Depth Lower Bounds: The XOR-KRW Conjecture}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{38:1--38:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14312},
  URN =		{urn:nbn:de:0030-drops-143121},
  doi =		{10.4230/LIPIcs.CCC.2021.38},
  annote =	{Keywords: communication complexity, KRW conjecture, circuit complexity, half-duplex communication complexity, Karchmer-Wigderson games, multiplexer relation, universal relation}
}

Keywords: communication complexity, KRW conjecture, circuit complexity, half-duplex communication complexity, Karchmer-Wigderson games, multiplexer relation, universal relation
Collection: 36th Computational Complexity Conference (CCC 2021)
Issue Date: 2021
Date of publication: 08.07.2021

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