Matching is as easy as matrix inversion
Combinatorica
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
The computational complexity of some problems of linear algebra
Journal of Computer and System Sciences
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
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
An algebraic approach to network coding
IEEE/ACM Transactions on Networking (TON)
Complexity classification of network information flow problems
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Maximum Matchings via Gaussian Elimination
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Dynamic Transitive Closure via Dynamic Matrix Inverse (Extended Abstract)
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
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
Graph Theory With Applications
Graph Theory With Applications
Matrices and Matroids for Systems Analysis
Matrices and Matroids for Systems Analysis
On the complexity of mixed discriminants and related problems
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
Polynomial time algorithms for multicast network code construction
IEEE Transactions on Information Theory
Computers & Mathematics with Applications
Deterministic Polynomial Time Algorithms for Matrix Completion Problems
SIAM Journal on Computing
A modified parallel optimization system for updating large-size time-evolving flow matrix
Information Sciences: an International Journal
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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Given a matrix whose entries are a mixture of numeric values and symbolic variables, the matrix completion problem is to assign values to the variables so as to maximize the resulting matrix rank. This problem has deep connections to computational complexity and numerous important algorithmic applications. Determining the complexity of this problem is a fundamental open question in computational complexity. Under different settings of parameters, the problem is variously in P, in RP, or NP-hard. We shed new light on this landscape by demonstrating a new region of NP-hard scenarios. As a special case, we obtain the first known hardness result for matrices in which each variable appears only twice.Another particular scenario that we consider is the simultaneous matrix completion problem, where one must simultaneously maximize the rank for several matrices that share variables. This problem has important applications in the field of network coding. Recent work has given a simple, greedy, deterministic algorithm for this problem, assuming that the algorithm works over a sufficiently large field. We show an exact threshold for the field size required to find a simultaneous completion efficiently. This result implies that, surprisingly, the simple greedy algorithm is optimal: finding a simultaneous completion over any smaller field is NP-hard.