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


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.

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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)


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
  • communication complexity
  • KRW conjecture
  • circuit complexity
  • half-duplex communication complexity
  • Karchmer-Wigderson games
  • multiplexer relation
  • universal relation


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