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Toward Better Depth Lower Bounds: The XOR-KRW Conjecture

Authors Ivan Mihajlin, Alexander Smal

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Ivan Mihajlin
  • St. Petersburg Department of Steklov Mathematical Institute of, Russian Academy of Sciences, Russia
Alexander Smal
  • St. Petersburg Department of Steklov Mathematical Institute of, Russian Academy of Sciences, Russia

Acknowledgements

We would like to thank the anonymous reviewers who have done a tremendous job carefully reading our paper and whose detailed comments helped us significantly improve the text of the paper and make it more readable.

Cite As Get BibTex

Ivan Mihajlin and Alexander Smal. Toward Better Depth Lower Bounds: The XOR-KRW Conjecture. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 38:1-38:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CCC.2021.38

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 𝐏 ̸ ⊆ 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • communication complexity
  • KRW conjecture
  • circuit complexity
  • half-duplex communication complexity
  • Karchmer-Wigderson games
  • multiplexer relation
  • universal relation

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References

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