Deterministic Polynomial Time Algorithms for Matrix Completion Problems

We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e. the problem of assigning values to the variables in a given symbolic matrix as to maximize the resulting matrix rank. Matrix completion belongs to the fundamental problems in computational complexity with numerous important algorithmic applications, among others, in computing dynamic transitive closures or multicast network codings (Harvey et al SODA 2005, Harvey et al SODA 2006). We design efficient deterministic algorithms for common generalizations of the results of Lovasz and Geelen on this problem by allowing linear functions in the entries of the input matrix such that the submatrices corresponding to each variable have rank one. We present also a deterministic polynomial time algorithm for finding the minimal number of generators of a given module structure given by matrices. We establish further several hardness results related to matrix algebras and modules. As a result we connect the classical problem of polynomial identity testing with checking surjectivity (or injectivity) between two given modules. One of the elements of our algorithm is a construction of a greedy algorithm for finding a maximum rank element in the more general setting of the problem. The proof methods used in this paper could be also of independent interest.Comment: 14 pages, preliminar

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

Paved with Good Intentions: Analysis of a Randomized Block Kaczmarz Method

The block Kaczmarz method is an iterative scheme for solving overdetermined least-squares problems. At each step, the algorithm projects the current iterate onto the solution space of a subset of the constraints. This paper describes a block Kaczmarz algorithm that uses a randomized control scheme to choose the subset at each step. This algorithm is the first block Kaczmarz method with an (expected) linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and its submatrices. The analysis reveals that the algorithm is most effective when it is given a good row paving of the matrix, a partition of the rows into well-conditioned blocks. The operator theory literature provides detailed information about the existence and construction of good row pavings. Together, these results yield an efficient block Kaczmarz scheme that applies to many overdetermined least-squares problem