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On Sunflowers and Matrix Multiplication 2012 IEEE 27th Conference on Computational Complexity On Sunflowers and Matrix Multiplication Noga Alon Amir Shpilka Christopher Umans Sackler School of Mathematics and Faculty of Computer Science Computing and Mathematical Sciences Blavatnik School of Computer Science Technion-Israel Institute of Technology California Institute of Technology Tel Aviv University Haifa, Israel Pasadena, CA 91125 Tel Aviv 69978, Israel Email: [email protected] Email: [email protected] and Institute for Advanced Study Princeton, New Jersey, 08540 Email: [email protected] Abstract—We present several variants of the sunflower |F| > (k − 1)s · s! then F contains a k-sunflower. conjecture of Erdos˝ and Rado [ER60] and discuss the relations They conjectured that actually many fewer sets are among them. needed. We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Winograd Conjecture 1 (Classical sunflower conjecture [ER60]): [CW90] and Cohn et al [CKSU05] regarding possible ap- For every k>0 there exists a constant ck such that the proaches for obtaining fast matrix multiplication algorithms. following holds. Let F be an arbitrary family of sets of Specifically, we show that the Erdos-Rado˝ sunflower conjec- size s from some universe U.If|F| ≥ cs then F contains ture (if true) implies a negative answer to the “no three k disjoint equivoluminous subsets” question of Coppersmith a k-sunflower. and Winograd [CW90]; we also formulate a “multicolored” This (and in particular the case k =3) is one of the most n sunflower conjecture in Z3 and show that (if true) it implies a well known open problems in combinatorics and despite negative answer to the “strong USP” conjecture of [CKSU05] a lot of attention it is still open today (see e.g. [Erd71], (although it does not seem to impact a second conjecture [Erd75], [Erd81]). in [CKSU05] 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 This conjecture has applications in combinatorial number et al. conjecture. theory and the study of Turan´ type problems in extremal Zn The multicolored sunflower conjecture in 3 is a strength- graph theory ([Fu91]), as well as in other areas in combina- ening of the well-known (ordinary) sunflower conjecture in n torics including the investigation of explicit constructions of Z3 , and we show via our connection that a construction from [CKSU05] yields a lower bound of (2.51 ...)n on the size of Ramsey graphs ([AP01]). A close variant has been applied the largest multicolored 3-sunflower-free set, which beats the in Circuit Complexity ([Ju01], see also [Ra85], [AB87]). current best known lower bound of (2.21 ...)n [Edel04] on the n size of the largest 3-sunflower-free set in Z3 . B. Matrix multiplication and conjectures implying an Keywords-Matrix Multiplication, Sunflower Conjecture O(n2+) time algorithm I. INTRODUCTION A fundamental algorithmic problem in computer science asks to compute the product of two given n × n matrices. A. Sunflowers This problem received a lot of attention since the seminal A k-sunflower (also called a Δ-system of size k)isa work of Strassen [Str69] that showed that one can do better collection of k sets, from some universe U, that have the than the simple row-column multiplication. For many years, same pairwise intersections. This notion was first introduced the best known result was due to Coppersmith and Wino- in [ER60] and proved itself to be a very useful tool in grad, who gave an O(n2.376...) time algorithm for multiply- combinatorics, number theory and computer science ever ing two n × n matrices [CW90]. Very recently, this running since. See, e.g., [Fu91], [Ju01], [AP01]. time has been lowered, by Stothers and Williams [S10], A basic problem concerning sunflowers is how many [W11], and the world record now stands at O(n2.373...) sets do we need in order to guarantee the existence of a [W11]. It is widely believed by experts that one should k-sunflower. Erdos˝ and Rado proved the following bound be able to multiply n × n matrices in time O(n2+), for [ER60], [ER69]. every >0 (see e.g., [Gat88], [BCS97]). It is a major open Theorem 1.1 (Erdos˝ and Rado [ER60]): Let F be an ar- problem to achieve such an algorithm (which we term fast bitrary family of sets of size s from some universe U.If matrix multiplication). 1093-0159/12 $26.00 © 2012 IEEE 214 DOI 10.1109/CCC.2012.26 In their paper, Coppersmith and Winograd proposed an We emphasize that the message of this paper with respect to approach towards achieving fast matrix multiplication. They approaches to matrix multiplication is not entirely negative. showed that the existence of an Abelian group and a In particular, not all of the statements labeled “conjectures” subset of it that satisfy certain conditions, imply that their in this paper have equal weight. After laying out the various techniques can yield an O(n2+) time algorithm. connections, we speculate on the relative likelihood of the A new approach for matrix multiplication that is based various conjectures, in Section IV. on group representation theory was suggested by Cohn and Umans [CU03]. In a subsequent work [CKSU05], Cohn D. Organization et al. gave several algorithms based on the framework We organize the paper as follows. In Section II we discuss of [CU03], and were even able to match the result of different sunflower conjectures and give the relations among [CW90]. Similarly to Coppersmith and Winograd, Cohn et them. In Section III we present the questions raised in al. formulated two conjectures regarding the existence of [CW90], [CKSU05] and show their relation to sunflowers. certain combinatorial structures that, if true, would yield In Section IV we discuss these implications. O(n2+) time matrix-multiplication algorithms. E. Notation C. Our results Z { − } We relate variants of the sunflower conjecture to each For an integer m we denote by m = 0,...,m 1 other, and to two of the aforementioned matrix multiplication the additive group modulo m. [n] stands for the set [n]= { } conjectures. Our main results are as follows: 1,...,n . All logarithms are taken in base 2. We will often use direct products of families of sets. For families F ⊂ • We prove (Theorem 3.2) that if Conjecture 1 is 1 Zn1 and F ⊂ Zn2 , we define their direct product to be the true then no Abelian group and subset satisfying the m 2 m family F ×F = {u◦v | u ∈F and v ∈F }, where u◦v Coppersmith-Winograd conjecture exists. Since Con- 1 2 1 2 is the concatenation of u and v,overZn1+n2 . When each F jecture 1 is well-known and widely believed, this is a m i is a family of subsets from some universe U , we define the strong indication that the Coppersmith-Winograd con- i direct product analogously, over the disjoint union U U . jecture may be false. 1 2 • We formulate a new variant of the sunflower conjecture II. SUNFLOWER CONJECTURES (Conjecture 6) and prove (Theorem 3.7) that it contra- dicts one of two conjectures in [CKSU05], regarding Definition 2.1: We say that k subsets A1,...,Ak,ofa ∀ ∩ ∩k the existence of “Strong Uniquely Solvable Puzzles.” universe U, form a k-sunflower if i = jAi Aj = =1A. Thus any construction of Strong Uniquely-Solvable Conjecture 1 states that for every k there is an integer s Puzzles that would yield an exponent 2 algorithm for ck such that any ck sets of size s contain a k-sunflower. matrix multiplication must disprove this variant of the The conjecture is open even for k =3, which is the case sunflower conjecture, which helps explain the difficulty that we are most interested in, in this paper. This case is of the problem. We stress though, that this does not rule also the main one studied in most existing papers on the out the possibility of obtaining fast matrix multiplica- conjecture, and it is believed that any proof of the conjecture tion algorithms using the Cohn-Umans framework. In for k =3is likely to provide a proof of the general particular, in [CKSU05] Cohn et al. propose a second case as well. Currently the best known result is given in direction that seems not to contradict the variants of the the following theorem of Kostochka, improving an earlier sunflower conjecture that are considered here. estimate of [Spe77]. • We discover via connections proved in this paper, that Theorem 2.2 (Kostochka [Ko97]): There exists a con- the Coppersmith-Winograd conjecture actually implies stant c such that every family of s-sets of size at least · log log log s s the Strong Uniquely-Solvable Puzzles conjecture of cs! ( log log s ) contains a 3-sunflower. [CKSU05]. Thus the latter conjecture is formally easier The following conjecture of Erdos˝ and Szemeredi´ [ES78] to prove. concerns 3-sunflowers inside [n]. • We show via connections established in this paper that Conjecture 2 (Sunflower conjecture in {0, 1}n [ES78]): a construction from [CKSU05] yields a lower bound There exists >0 such that any family F of subsets of [n] of (2.51 ...)n on the size of the largest multicolored 3- (n ≥ 2) of size |F| ≥ 2(1−)·n contains a 3-sunflower. sunflower-free set, which beats the current best known Note the difference between Conjectures 1 and 2. While lower bound of (2.21 ...)n [Edel04] on the size of the Conjecture 1 concerns s-sets from an unbounded universe, Zn largest 3-sunflower-free set in 3 .
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