Multiplication is one of the most fundamental computational problems, yet its
true complexity remains elusive. The best known upper bound, by Fürer,
shows that two $n$-bit numbers can be multiplied via a boolean circuit of size
$O(n n 4^łg^*n)$, where $łg^*n$ is the very slowly growing
iterated logarithm. In this work, we prove that if a central conjecture in the
area of network coding is true, then any constant degree boolean circuit for
multiplication must have size $Ømega(n n)$, thus almost completely
settling the complexity of multiplication circuits. We additionally revisit
classic conjectures in circuit complexity, due to Valiant, and show that the
network coding conjecture also implies one of Valiant's conjectures.
%0 Journal Article
%1 afshani2019lower
%A Afshani, Peyman
%A Freksen, Casper Benjamin
%A Kamma, Lior
%A Larsen, Kasper Green
%D 2019
%K bounds gates
%T Lower Bounds for Multiplication via Network Coding
%U http://arxiv.org/abs/1902.10935
%X Multiplication is one of the most fundamental computational problems, yet its
true complexity remains elusive. The best known upper bound, by Fürer,
shows that two $n$-bit numbers can be multiplied via a boolean circuit of size
$O(n n 4^łg^*n)$, where $łg^*n$ is the very slowly growing
iterated logarithm. In this work, we prove that if a central conjecture in the
area of network coding is true, then any constant degree boolean circuit for
multiplication must have size $Ømega(n n)$, thus almost completely
settling the complexity of multiplication circuits. We additionally revisit
classic conjectures in circuit complexity, due to Valiant, and show that the
network coding conjecture also implies one of Valiant's conjectures.
@article{afshani2019lower,
abstract = {Multiplication is one of the most fundamental computational problems, yet its
true complexity remains elusive. The best known upper bound, by F\"{u}rer,
shows that two $n$-bit numbers can be multiplied via a boolean circuit of size
$O(n \lg n \cdot 4^{\lg^*n})$, where $\lg^*n$ is the very slowly growing
iterated logarithm. In this work, we prove that if a central conjecture in the
area of network coding is true, then any constant degree boolean circuit for
multiplication must have size $\Omega(n \lg n)$, thus almost completely
settling the complexity of multiplication circuits. We additionally revisit
classic conjectures in circuit complexity, due to Valiant, and show that the
network coding conjecture also implies one of Valiant's conjectures.},
added-at = {2019-03-03T07:45:20.000+0100},
author = {Afshani, Peyman and Freksen, Casper Benjamin and Kamma, Lior and Larsen, Kasper Green},
biburl = {https://www.bibsonomy.org/bibtex/243e43f8a0651c3ac54292aca99df29af/kirk86},
description = {1902.10935.pdf},
interhash = {007533b14b8d65928a41e940270992c7},
intrahash = {43e43f8a0651c3ac54292aca99df29af},
keywords = {bounds gates},
note = {cite arxiv:1902.10935},
timestamp = {2019-03-03T07:45:20.000+0100},
title = {Lower Bounds for Multiplication via Network Coding},
url = {http://arxiv.org/abs/1902.10935},
year = 2019
}