computational complexity

, Volume 22, Issue 2, pp 219–243 | Cite as

On sunflowers and matrix multiplication

Article

Abstract

We present several variants of the sunflower conjecture of Erdős & 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 Erdős–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 \({\mathbb{Z}_3^n}\) 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 \({\mathbb{Z}_3^n}\) is a strengthening of the well-known (ordinary) sunflower conjecture in \({\mathbb{Z}_3^n}\) , 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 \({\mathbb{Z}_3^n}\) .

Keywords

sunflowers matrix multiplication 

Subject classification

05D05 68Q25 

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References

  1. 1.
    Alon N., Boppana R.B (1987) The monotone circuit complexity of Boolean functions. Combinatorica 7(1): 1–22MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Zs. Baranyai (1975). On the factorization of the complete uniform hypergraph. In Infinite and finite sets, Proc. Coll. Keszthely, 1973, (A. Hajnal, R. Rado and V.T. Sós, eds.), Colloquia Math. Soc. János Bolyai 10. North-Holland.Google Scholar
  3. 3.
    Bateman M., Katz N.H (2012) New bounds on cap sets. J. of the AMS 25(2): 585–613MathSciNetMATHGoogle Scholar
  4. 4.
    P. Bürgisser, M. Clausen & M. A. Shokrollahi (1997). Algebraic Complexity Theory. Springer.Google Scholar
  5. 5.
    H. Cohn, R. D. Kleinberg, B. Szegedy & C. Umans (2005). Group-theoretic Algorithms for Matrix Multiplication. In Proceedings of the 46th Annual FOCS, 379–388.Google Scholar
  6. 6.
    H. Cohn & C. Umans (2003). A Group-theoretic Approach to Fast Matrix Multiplication. In Proceedings of the 44th Annual FOCS, 438–449.Google Scholar
  7. 7.
    D. Coppersmith & S. Winograd (1990). Matrix multiplication via arithmetic progression. J. of Symbolic Computation 9, 251–280.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    W. A. Deuber, P. Erdös, D. S. Gunderson, A. V. Kostochka & A. G. Meyer (1997). Intersection Statements for Systems of Sets. J. Comb. Theory, Ser. A 79(1), 118–132.Google Scholar
  9. 9.
    Y. Edel (2004). Extensions of generalized product caps 31(1), 5–14.Google Scholar
  10. 10.
    P. Erdős (1971). Some unsolved problems in graph theory and com binatorial analysis. In Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), 97–109. Academic Press, London.Google Scholar
  11. 11.
    P. Erdős (1975). Problems and results on finite and infinite combi natorial analysis. In Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdös on his 60th birthday) Vol. I; Colloq. Math. Soc., János Bolyai, editor, volume 10, 403–424. North-Holland, Amster717 dam.Google Scholar
  12. 12.
    P. Erdős (1981). On the combinatorial problems which I would most like to see solved. Combinatorica 1(1), 25–42.Google Scholar
  13. 13.
    P. Erdős & R. Rado (1960). Intersection theorems for systems of sets. J. London Math. Soc. 35, 85–90.Google Scholar
  14. 14.
    P. Erdős & R. Rado (1969). Intersection theorems for systems of sets II. J. London Math. Soc. 44, 467–479.Google Scholar
  15. 15.
    P. Erdős & E. Szemerédi (1978). Combinatorial properties of sys tems of sets. J. Combinatorial Theory Ser. A 24(3), 308–313.Google Scholar
  16. 16.
    Z. Füredi (1991). Turán type problems. Surveys in combinatorics, London Math. Soc. Lecture Note Ser. 166, 253–300. Cambridge Univ. Press.Google Scholar
  17. 17.
    J. von zur Gathen (1988). Algebraic Complexity Theory. Annual review of computer science 3, 317–347.CrossRefGoogle Scholar
  18. 18.
    S. Jukna (2001). Extremal Combinatorics. Springer.Google Scholar
  19. 19.
    A. V. Kostochka (1997). A bound of the cardinality of families not containing Δ-systems. In The mathematics of Paul Erdős, II, Algo rithms Combin., volume 14, 229–235. Springer.Google Scholar
  20. 20.
    R. Meshulam (1995). On Subsets of Finite Abelian Groups with No 3-Term Arithmetic Progressions. J. Comb. Theory, Ser. A 71(1), 168–172.Google Scholar
  21. 21.
    A. A. Razborov (1985). Lower bounds on the monotone complexity of some Boolean functions. Dokl. Akad. Nauk. SSSR 281(4), 798–801. In Russian.Google Scholar
  22. 22.
    R. Salem & D. Spencer (1942). On sets of integers which contain no three in arithmetic progression. Proc. Nat. Acad. Sci. (USA) 28, 561.Google Scholar
  23. 23.
    J. Spencer (1977). Intersection Theorems for Systems of Sets. Canadian Math Bulletin 20(2), 249–254.MATHCrossRefGoogle Scholar
  24. 24.
    A. Stothers (2010). On the complexity of matrix multiplication. Ph.D. thesis, U. Edinburgh.Google Scholar
  25. 25.
    Strassen V (1969) Gaussian elimination is not optimal. Numer. Math 13: 354–356MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    V. Strassen (1987). Relative bilinear complexity and matrix multi plication.J. Reine Angew. Math. 375/376, 406–443.Google Scholar
  27. 27.
    V. Vassilevska Williams (2012). Multiplying matrices faster than Coppersmith-Winograd. In Proceedings of the 44th annual STOC, 887–898.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Sackler School of Mathematics and Blavatnik School of Computer ScienceTel Aviv UnivetyTel AvivIsrael
  2. 2.Institute for Advanced StudyPrincetonUSA
  3. 3.Faculty of Computer ScienceTechnion-Israel Institute of TechnologyHaifaIsrael
  4. 4.Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA

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