Title: The Hardness of the Noncommutative Determinant
Speaker: Srikanth Srinivasan, The Institute of Mathematical Sciences, Chennai
Date: Wednesday, April 13, 2011 11:00-12:00pm
Location: DIMACS Center, CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ
We study the complexity of computing the determinant of a matrix over a non-commutative 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 hardness and easiness of noncommutative determinant computation.
* [Hardness] Computing the determinant of an n \times n matrix whose entries are themselves 2 \times 2 matrices over a field is as hard as computing the permanent over the field. This extends the recent result of Arvind and Srinivasan, who proved a similar result which however required the entries to be of linear dimension.
* [Easiness] Determinant of an n \times n matrix whose entries are themselves d \times d upper triangular matrices can be computed in poly(n^d) time. Combining the above with the decomposition theorem of finite dimensional algebras (in particular exploiting the simple structure of 2 \times 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 non-commutative, 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., the smallest d such that R(A)^d = 0), then there exists a poly(n^d)-time algorithm that computes determinants over the algebra A.
In particular, for any constant dimensional algebra A over a finite field, 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 feasible and otherwise determinant is as hard as permanent.
Joint work with Steve Chien, Prahladh Harsha, and Alistair Sinclair.