Almost settling the hardness of noncommutative determinant

  • Authors:

  • Steve Chien;Prahladh Harsha;Alistair Sinclair;Srikanth Srinivasan

  • Affiliations:

  • Google, Mountain View, CA, USA;TIFR, Mumbai, India;University of California Berkeley, Berkeley, CA, USA;Institute for Advanced Study, Princeton, NJ, USA

  • Venue:
  • Proceedings of the forty-third annual ACM symposium on Theory of computing
  • Year:
  • 2011

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Abstract

In this paper, we study the complexity of computing the determinant of a matrix over a noncommutative algebra. In particular, we ask the question: "Over which algebras is the determinant easier to compute than the permanent?" Towards resolving this question, we show the following results for noncommutative determinant computation: [Hardness] Computing the determinant of an n x n matrix whose entries are themselves 2 x 2 matrices over any field of zero or odd characteristic is as hard as computing the permanent over the field. This extends the recent result of Arvind and Srinivasan, which required the entries to be matrices of dimension linear in n. [Easiness] The determinant of an n x n matrix whose entries are themselves d x d upper triangular matrices can be computed in poly(nd) time. Combining the above with the decomposition theorem for finite dimensional algebras (and in particular exploiting the simple structure of 2 x 2 matrix algebras), we can extend the above hardness and easiness statements to more general algebras as follows. Let A be a finite dimensional algebra over a finite field of odd characteristic with radical R(A). [Hardness] If the quotient A/R(A) is noncommutative, then computing the determinant over the algebra A is as hard as computing the permanent. [Easiness] If the quotient A/R(A) is commutative, and furthermore R(A) has nilpotency index d (i.e., d is the smallest integer such that R(A)d =0), then there exists a poly(nd)-time algorithm that computes determinants over the algebra A. In particular, for any constant dimensional algebra A over a finite field of odd characteristic, since the nilpotency index of R(A) is at most a constant, we have the following dichotomy theorem: if A/R(A) is commutative then efficient determinant computation is possible, and otherwise determinant is as hard as permanent.