Journal of Symbolic Computation
Polynomial time algorithms for modules over finite dimensional algebras
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Mathematics of Operations Research
Random Structures & Algorithms
A deterministic polynomial-time algorithm for approximating mixed discriminant and mixed volume
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
A Linear Lower Bound on the Unbounded Error Probabilistic Communication Complexity
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Matrix integrals and map enumeration: An accessible introduction
Mathematical and Computer Modelling: An International Journal
Further Results on the Cross Norm Criterion for Separability
Quantum Information Processing
The computational complexity of linear optics
Proceedings of the forty-third annual ACM symposium on Theory of computing
A quasipolynomial-time algorithm for the quantum separability problem
Proceedings of the forty-third annual ACM symposium on Theory of computing
Quantum Information & Computation
Deterministic Polynomial Time Algorithms for Matrix Completion Problems
SIAM Journal on Computing
On the complexity of mixed discriminants and related problems
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
Epsilon-net method for optimizations over separable states
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
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Generalizing a decision problem for bipartite perfect matching, Edmonds (J. Res. Natl. Bur. Standards 718(4) (1967) 242) introduced the problem (now known as the Edmonds Problem) of deciding if a given linear subspace of M(N) contains a non-singular matrix, where M(N) stands for the linear space of complex N × N matrices. This problem led to many fundamental developments in matroid theory, etc.Classical matching theory can be defined in terms of matrices with non-negative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with non-negative entries. (Here operator refers to maps from matrices to matrices.) First, we reformulate the Edmonds Problem in terms of completely positive operators, or equivalently, in terms of bipartite density matrices. It turns out that one of the most important cases when Edmonds' problem can be solved in polynomial deterministic time, i.e. an intersection of two geometric matroids, corresponds to unentangled (aka separable) bipartite density matrices. We introduce a very general class (or promise) of linear subspaces of M(N) on which there exists a polynomial deterministic time algorithm to solve Edmonds' problem. The algorithm is a thoroughgoing generalization of algorithms in Linial, Samorodnitsky and Wigderson, Proceedings of the 30th ACM Symposium on Theory of Computing, ACM, New York, 1998; Gurvits and Yianilos, and its analysis benefits from an operator analog of permanents, so-called Quantum Permanents.Finally, we prove that the weak membership problem for the convex set of separable normalized bipartite density matrices is NP-HARD.