Fooling-sets and rank in nonzero characteristic

  • Mirjam Friesen
  • Dirk Oliver Theis
Part of the CRM Series book series (PSNS, volume 16)

Abstract

An n × n matrix M is called a fooling-set matrix of size n, if its diagonal entries are nonzero, whereas for every k ≠ ℓ we have Mk,ℓMℓ,k = 0. Dietzfel-binger, Hromkovič, and Schnitger (1996) showed that n ≤ (rkM)2, regardless of over which field the rank is computed, and asked whether the exponent on rkM can be improved.

We settle this question for nonzero characteristic by constructing a family of matrices for which the bound is asymptotically tight. The construction uses linear recurring sequences.

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References

  1. [1]
    S. Arora and B. Barak, “Computational Complexity”, Cambridge University Press, Cambridge, 2009, A modern approach. MR 2500087 (2010i:68001) 2Google Scholar
  2. [2]
    J. E. Cohen and U. G. Rothblum, Nonnegative ranks, decompositions, and factorizations of nonnegative matrices, Linear Algebra Appl. 190 (1993), 149–168. MR 1230356 (94i: 15015) 3CrossRefMathSciNetGoogle Scholar
  3. [3]
    M. Dawande, A notion of cross-perfect bipartite graphs, Inform. Process. Lett. 88(4) (2003), 143–147. MR 2009283 (2004g:05118) 3CrossRefMathSciNetGoogle Scholar
  4. [4]
    M. Dietzfelbinger, J. Hromkovič and G. Schnitger, A comparison of two lower-bound methods for communication complexity, Theoret. Comput. Sci. 168(1) (1996), 39–51, 19th International Symposium on Mathematical Foundations of Computer Science (Košice, 1994). MR 1424992 (98a:68068) 1, 2CrossRefMathSciNetGoogle Scholar
  5. [5]
    S. Fiorini, V. Kaibel, K. Pashkovich and D. O. Theis, Combinatorial bounds on nonnegative rank and extended formulations, http://arxiv.org/abs/1111.0444, arXiv:1111.0444) Discrete Math. (2013), to appear.
  6. [6]
    H. Gruber and M. Holzer, Finding lower bounds for nonde-terministic state complexity is hard (extended abstract), Developments in language theory, Lecture Notes in Comput. Sci., Vol. 4036, Springer, Berlin, 2006, pp. 363–374. MR 2334484 2Google Scholar
  7. [7]
    S. Jukna and A. S. Kulikov, On covering graphs by complete bipartite subgraphs, Discrete Math. 309(10) (2009), 3399–3403. MR 2526759 (2010h:05231) 3CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    H. Klauck and R. de Wolf, Fooling one-sided quantum protocols, http://arxiv.org/abs/1204.4619, arXiv: 1204.4619, 2012. 1,2,5
  9. [9]
    S. Kopparty and K. P. S. Bhaskara Rao, The minimum rank problem: a counterexample, Linear Algebra Appl. 428(7) (2008), 1761–1765. MR 2388655 (2009a: 15002) 6CrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    E. Kushilevitz and N. Nisan, “Communication Complexity”, Cambridge University Press, Cambridge, 1997. MR 1426129 (98c:68074) 1, 2Google Scholar
  11. [11]
    R. Lidl and H. Niederreiter, “Introduction to Finite Fields and their Applications”, first ed., Cambridge University Press, Cambridge, 1994. MR 1294139 (95f: 11098) 5Google Scholar
  12. [12]
    L. Lovász and M. Saks, Möbius functions and communication complexity, Proc. 29th IEEE FOCS, IEEE, 1988, 81–90.Google Scholar
  13. [13]
    J. A. Soto and C. Telha, Jump number of two-directional orthogonal ray graphs, Integer programming and combinatorial optimization, Lecture Notes in Comput. Sci., vol. 6655, Springer, Heidelberg, 2011, 389–403. MR 2820923 (2012j:05305) 3Google Scholar
  14. [14]
    M. Yannakakis, Expressing combinatorial optimization problems by linear programs, J. Comput. System Sci. 43(3) (1991), 441–466. MR 1135472 (93a:90054) 2CrossRefMathSciNetGoogle Scholar

Copyright information

© Scuola Normale Superiore Pisa 2013

Authors and Affiliations

  • Mirjam Friesen
    • 1
  • Dirk Oliver Theis
    • 2
  1. 1.Faculty of MathematicsOtto von Guericke University MagdeburgGermany
  2. 2.Faculty of Mathematics and Computer ScienceUniversity of TartuEstonia

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