Hardness Condensation and Log Approximate Rank Conjecture


Artur Riazanov October 20, 2023.


Can a function f: {0,1}^n -> {0,1} of query complexity k « n be restricted to O(k) variables such that its query complexity remains to be Omega(k)? That is, can query complexity be condensed via restriction? We study this question for query and communication complexity. On the negative side, we show that query complexity can not be condensed. In this talk, however, I will focus on a positive result in this direction: a function Sink o XOR that was used to falsify the log approximate rank conjecture (LARC) can be condensed in the sense above. The condensation is itself a stronger counterexample to LARC which was sought by Chattopadhyay, Garg, and Sherif (2021). Based on the joint work with Mika Göös, Ilan Newman, and Dmitry Sokolov.

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