Lower Bounds for Multiplication via Network Coding
Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, very recently proved by Harvey and van der Hoeven (2019), shows that two n-bit numbers can be multiplied via a boolean circuit of size O(n lg n). 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.
Circuit Complexity
Circuit Lower Bounds
Multiplication
Network Coding
Fine-Grained Complexity
Theory of computation
Theory of computation~Circuit complexity
10:1-10:12
Track A: Algorithms, Complexity and Games
Peyman Afshani is supported by DFF (Det Frie Forskningsraad) of Danish Council for Independent Research under grant ID DFF-7014-00404.
Casper Benjamin Freksen and Lior Kamma are supported by Villum Young Investigator Grant 13163.
Kasper Green Larsen is supported by Villum Young Investigator Grant 13163 and an AUFF Starting Grant.
Peyman
Afshani
Peyman Afshani
Computer Science Department, Aarhus University, Denmark
Casper Benjamin
Freksen
Casper Benjamin Freksen
Computer Science Department, Aarhus University, Denmark
Lior
Kamma
Lior Kamma
Computer Science Department, Aarhus University, Denmark
Kasper Green
Larsen
Kasper Green Larsen
Computer Science Department, Aarhus University, Denmark
10.4230/LIPIcs.ICALP.2019.10
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Peyman Afshani, Casper Freksen, Lior Kamma, and Kasper G. Larsen
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