On sunflowers and matrix multiplication

Noga Alon, Amir Shpilka, Christopher Umans

Research output: Contribution to journalArticle

23 Scopus citations

Abstract

We present several variants of the sunflower conjecture of Erdo{double acute}s & Rado (J Lond Math Soc 35:85-90, 1960) and discuss the relations among them.We then show that two of these conjectures (if true) imply negative answers to the questions of Coppersmith & Winograd (J Symb Comput 9:251-280, 1990) and Cohn et al. (2005) regarding possible approaches for obtaining fast matrix-multiplication algorithms. Specifically, we show that the Erdo{double acute}s-Rado sunflower conjecture (if true) implies a negative answer to the "no three disjoint equivoluminous subsets" question of Coppersmith & Winograd (J Symb Comput 9:251-280, 1990); we also formulate a "multicolored" sunflower conjecture in ℤ3n and show that (if true) it implies a negative answer to the "strong USP" conjecture of Cohn et al. (2005) (although it does not seem to impact a second conjecture in Cohn et al. (2005) or the viability of the general group-theoretic approach). A surprising consequence of our results is that the Coppersmith-Winograd conjecture actually implies the Cohn et al. conjecture.The multicolored sunflower conjecture in ℤ3n is a strengthening of the well-known (ordinary) sunflower conjecture in ℤ3n, and we show via our connection that a construction from Cohn et al. (2005) yields a lower bound of (2.51 . . .)n on the size of the largest multicolored 3-sunflower-free set, which beats the current best-known lower bound of (2.21 . . .) n Edel (2004) on the size of the largest 3-sunflower-free set in ℤ3n.

Original languageEnglish (US)
Pages (from-to)219-243
Number of pages25
JournalComputational Complexity
Volume22
Issue number2
DOIs
StatePublished - Jun 2013

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Mathematics(all)
  • Computational Theory and Mathematics
  • Computational Mathematics

Keywords

  • matrix multiplication
  • sunflowers

Fingerprint Dive into the research topics of 'On sunflowers and matrix multiplication'. Together they form a unique fingerprint.

  • Cite this