On sunflowers and matrix multiplication

Noga Alon, Amir Shpilka, Christopher Umans

Research output: Chapter in Book/Report/Conference proceedingConference contribution

5 Scopus citations

Abstract

We present several variants of the sunflower conjecture of Erdos and Rado [ER60] and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Wino grad [CW90] and Cohn et al [CKSU05] regarding possible approaches for obtaining fast matrix multiplication algorithms. Specifically, we show that the Erdos-Rado sunflower conjecture (if true) implies a negative answer to the ''no three disjoint equivoluminous subsets'' question of Coppersmith and Wino grad [CW90]; we also formulate a ''multicolored'' sunflower conjecture in Z-3 n and show that (if true) it implies a negative answer to the ''strong USP'' conjecture of [CKSU05] (although it does not seem to impact a second conjecture in [CKSU05] or the viability of the general group-theoretic approach). A surprising consequence of our results is that the Coppersmith-Wino grad conjecture actually implies the Cohn et al. conjecture. The multicolored sunflower conjecture in Z-3 n is a strengthening of the well-known (ordinary) sunflower conjecture in Z-3 n, and we show via our connection that a construction from [CKSU05] yields a lower bound of (2.51\ldots) n on the size of the largest {\em multicolored} 3-sunflower-free set, which beats the current best known lower bound of (2.21\ldots) n [Edel04] on the size of the largest 3-sunflower-free set in Z-3 n.

Original languageEnglish (US)
Title of host publicationProceedings - 2012 IEEE 27th Conference on Computational Complexity, CCC 2012
Pages214-223
Number of pages10
DOIs
StatePublished - 2012
EventIEEE Computer Society Technical Committee on Mathematical Foundations of Computing - Porto, Portugal
Duration: Jun 26 2012Jun 29 2012

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)1093-0159

Other

OtherIEEE Computer Society Technical Committee on Mathematical Foundations of Computing
CountryPortugal
CityPorto
Period6/26/126/29/12

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Computational Mathematics

Keywords

  • Matrix Multiplication
  • Sunflower Conjecture

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    Alon, N., Shpilka, A., & Umans, C. (2012). On sunflowers and matrix multiplication. In Proceedings - 2012 IEEE 27th Conference on Computational Complexity, CCC 2012 (pp. 214-223). [6243397] (Proceedings of the Annual IEEE Conference on Computational Complexity). https://doi.org/10.1109/CCC.2012.26