Polynomial time solutions of some problems of computational algebra
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Polynomial time algorithms for modules over finite dimensional algebras
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Probabilistic Algorithms for Deciding Equivalence of Straight-Line Programs
Journal of the ACM (JACM)
The computational complexity of some problems of linear algebra
Journal of Computer and System Sciences
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Classical deterministic complexity of Edmonds' Problem and quantum entanglement
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Derandomizing polynomial identity tests means proving circuit lower bounds
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
The linear delta-matroid parity problem
Journal of Combinatorial Theory Series B
Classical complexity and quantum entanglement
Journal of Computer and System Sciences - Special issue: STOC 2003
Matroid Matching Via Mixed Skew-Symmetric Matrices
Combinatorica
Deterministic network coding by matrix completion
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
The complexity of matrix completion
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Matrices and Matroids for Systems Analysis
Matrices and Matroids for Systems Analysis
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Algebraic independence and blackbox identity testing
Information and Computation
Fast deterministic algorithms for matrix completion problems
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
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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 to maximize the resulting matrix rank. Matrix completion is one of the fundamental problems in computational complexity. It has numerous important algorithmic applications, among others, in computing dynamic transitive closures or multicast network codings [N. J. A. Harvey, D. R. Karger, and K. Murota, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2005, pp. 489-498; N. J. A. Harvey, D. R. Karger, and S. Yekhanin, Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2006, pp. 1103-1111]. We design efficient deterministic algorithms for common generalizations of the results of Lovász and Geelen on this problem by allowing linear polynomials in the entries of the input matrix such that the submatrices corresponding to each variable have rank one. Our methods are algebraic and quite different from those of Lovász and Geelen. We look at the problem of matrix completion in the more general setting of linear spaces of linear transformations and find a maximum rank element there using a greedy method. Matrix algebras and modules play a crucial role in the algorithm. We show (hardness) results for special instances of matrix completion naturally related to matrix algebras; i.e., in contrast to computing isomorphisms of modules (for which there is a known deterministic polynomial time algorithm), finding a surjective or an injective homomorphism between two given modules is as hard as the general matrix completion problem. The same hardness holds for finding a maximum dimension cyclic submodule (i.e., generated by a single element). For the “dual” task, i.e., finding the minimal number of generators of a given module, we present a deterministic polynomial time algorithm. The proof methods developed in this paper apply to fairly general modules and could also be of independent interest.