The Complexity of Surjective Homomorphism Problems -- a Survey

We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of which remain of open complexity

Almost Settling the Hardness of Noncommutative Determinant

In this paper, 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 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.Comment: 20 pages, 3 figure