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We design a deterministic polynomial time cn approximation algorithm for the permanent of positive semidefinite matrices where c = e+1 ⋍ 4:84. We write a natural convex relaxation and show that its optimum solution gives a cn approximation of the permanent. We further show that this factor is asymptotically tight by constructing a family of positive semidefinite matrices. We also show that...
Symbolic matrices in non-commuting variables, andthe related structural and algorithmic questions, have a remarkablenumber of diverse origins and motivations. They ariseindependently in (commutative) invariant theory and representationtheory, linear algebra, optimization, linear system theory,quantum information theory, and naturally in non-commutativealgebra.
The radio sky at frequencies below ∼30 MHz is virtually unobservable from Earth due to ionospheric disturbances and the opaqueness of the ionosphere below ∼10MHz, and also due to strong terrestrial radio interference. Deploying a radio observatory in space would open up this largely unexplored frequency band for science in astronomy, cosmology, geophysics, and space science. A Chinese-European team...
Motivated by questions in robust control and switched linear dynamical systems, we consider the problem checking whether every element of a polytope of n×n matrices A is stable. We show that this can be done in polynomial-time in n when the number of extreme points of A is constant, but becomes NP-Hard when the number of extreme points grows as Θ(n). This result has two useful corollaries: (i) for...
We give new lower and upper bounds on the permanent of a doubly stochastic matrix. Combined with previous work, this improves on the deterministic approximation factor. We also give a combinatorial application of the lower bound, proving S. Friedland's "Asymptotic Lower Matching Conjecture"for the monomer-dimer problem.
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