Hardness of Constant-Round Communication Complexity
How difficult is it to compute the communication complexity of a two-argument total Boolean function f:[N]×[N] → {0,1}, when it is given as an N×N binary matrix? In 2009, Kushilevitz and Weinreb showed that this problem is cryptographically hard, but it is still open whether it is NP-hard.
In this work, we show that it is NP-hard to approximate the size (number of leaves) of the smallest constant-round protocol for a two-argument total Boolean function f:[N]×[N] → {0,1}, when it is given as an N×N binary matrix. Along the way to proving this, we show a new deterministic variant of the round elimination lemma, which may be of independent interest.
NP-completeness
Communication Complexity
Round Elimination Lemma
Meta-Complexity
Theory of computation~Communication complexity
Theory of computation~Problems, reductions and completeness
31:1-31:30
Regular Paper
https://eccc.weizmann.ac.il/report/2021/030/
The authors would like to thank Ryan Williams for his support, and for several discussions and suggestions, without which this paper would not have existed. The authors would also like to thank Igor Oliveira for helpful conversations about hardness of communication complexity.
Shuichi
Hirahara
Shuichi Hirahara
National Institute of Informatics, Tokyo, Japan
This work was partly carried out during a visit supported by ACT-I, JST.
Rahul
Ilango
Rahul Ilango
Massachusetts Institute of Technology, Cambridge, MA, USA
During this work, this author was funded by an Akamai Presidential Fellowship and by NSF Grants CCF-1741615 and CCF-1909429.
Bruno
Loff
Bruno Loff
INESC-Tec and University of Porto, Portugal
This project was financed by the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia, within project UIDB/50014/2020. This work was partly carried out during a research visit conducted with support from DIMACS in association with its Special Focus on Lower Bounds.
10.4230/LIPIcs.CCC.2021.31
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