Improved construction for universality of determinant and permanent

  • Authors:

  • Hong Liu;Kenneth W. Regan

  • Affiliations:

  • Department of CSE, University at Buffalo, Bell Hall, Buffalo, NY;Department of CSE, University at Buffalo, Bell Hall, Buffalo, NY

  • Venue:
  • Information Processing Letters
  • Year:
  • 2006

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Abstract

Valiant [L. Valiant, Completeness classes in algebra, in: Proc. 11th Annual ACM Symposium on the Theory of Computing, Atlanta, GA, 1979, pp. 249-261] proved that every polynomial of formula size e is a projection of the (e + 2) × (e + 2) determinant polynomial. We improve "e + 2" to "e + 1", also for a definition of formula size that does not count multiplications by constants as gates. Our proof imitates the "2e + 2" proof of von zur Gathen [J. von zur Gathen, Feasible arithmetic computations: Valiant's hypothesis, Journal of Symbolic Computation 4 (1987) 137-172], but uses different invariants and a tighter set of base cases.