129 research outputs found
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
Arithmetic Circuits and the Hadamard Product of Polynomials
Motivated by the Hadamard product of matrices we define the Hadamard product
of multivariate polynomials and study its arithmetic circuit and branching
program complexity. We also give applications and connections to polynomial
identity testing. Our main results are the following. 1. We show that
noncommutative polynomial identity testing for algebraic branching programs
over rationals is complete for the logspace counting class \ceql, and over
fields of characteristic the problem is in \ModpL/\Poly. 2.We show an
exponential lower bound for expressing the Raz-Yehudayoff polynomial as the
Hadamard product of two monotone multilinear polynomials. In contrast the
Permanent can be expressed as the Hadamard product of two monotone multilinear
formulas of quadratic size.Comment: 20 page
On Explicit Branching Programs for the Rectangular Determinant and Permanent Polynomials
We study the arithmetic circuit complexity of some well-known family of polynomials through the lens of parameterized complexity. Our main focus is on the construction of explicit algebraic branching programs (ABP) for determinant and permanent polynomials of the rectangular symbolic matrix in both commutative and noncommutative settings. The main results are:
- We show an explicit O^*(binom{n}{downarrow k/2})-size ABP construction for noncommutative permanent polynomial of k x n symbolic matrix. We obtain this via an explicit ABP construction of size O^*(binom{n}{downarrow k/2}) for S_{n,k}^*, noncommutative symmetrized version of the elementary symmetric polynomial S_{n,k}.
- We obtain an explicit O^*(2^k)-size ABP construction for the commutative rectangular determinant polynomial of the k x n symbolic matrix.
- In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is #W[1]-hard
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
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