文章基本信息

标题：On Sunflowers and Matrix Multiplication
本地全文：下载
作者：Noga Alon ; Amir Shpilka ; Chris Umans 等
期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2011
卷号：2011
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：We present several variants of the sunflower conjecture of Erd\H{o}s and Rado and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Winograd and Cohn et al. regarding possible approaches for obtaining fast matrix multiplication algorithms. Specifically, we show that the Erd\H{o}s-Rado sunflower conjecture (if true) implies a negative answer to the ``no three disjoint equivoluminous subsets'' question of Coppersmith and Winograd; we also formulate a ``multicolored'' sunflower conjecture in Zn3 and show that (if true) it implies a negative answer to the ``strong USP'' conjecture of Cohn et al. (although it does not seem to impact a second conjecture in that paper 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 Zn3 is a strengthening of the well-known (ordinary) sunflower conjecture in Zn3, and we show via our connection that a construction of Cohn et al. yields a lower bound of (251)n on the size of the largest {\em multicolored} 3-sunflower-free set, which beats the current best known lower bound of (221)n on the size of the largest 3-sunflower-free set in Zn3.
关键词：matrix multiplication, Sunflowers