14,159 research outputs found
FPT-algorithms for some problems related to integer programming
In this paper, we present FPT-algorithms for special cases of the shortest
lattice vector, integer linear programming, and simplex width computation
problems, when matrices included in the problems' formulations are near square.
The parameter is the maximum absolute value of rank minors of the corresponding
matrices. Additionally, we present FPT-algorithms with respect to the same
parameter for the problems, when the matrices have no singular rank
sub-matrices.Comment: arXiv admin note: text overlap with arXiv:1710.00321 From author:
some minor corrections has been don
The Width and Integer Optimization on Simplices With Bounded Minors of the Constraint Matrices
In this paper, we will show that the width of simplices defined by systems of
linear inequalities can be computed in polynomial time if some minors of their
constraint matrices are bounded. Additionally, we present some
quasi-polynomial-time and polynomial-time algorithms to solve the integer
linear optimization problem defined on simplices minus all their integer
vertices assuming that some minors of the constraint matrices of the simplices
are bounded.Comment: 12 page
Randomized Dynamic Mode Decomposition
This paper presents a randomized algorithm for computing the near-optimal
low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging
techniques to compute low-rank matrix approximations at a fraction of the cost
of deterministic algorithms, easing the computational challenges arising in the
area of `big data'. The idea is to derive a small matrix from the
high-dimensional data, which is then used to efficiently compute the dynamic
modes and eigenvalues. The algorithm is presented in a modular probabilistic
framework, and the approximation quality can be controlled via oversampling and
power iterations. The effectiveness of the resulting randomized DMD algorithm
is demonstrated on several benchmark examples of increasing complexity,
providing an accurate and efficient approach to extract spatiotemporal coherent
structures from big data in a framework that scales with the intrinsic rank of
the data, rather than the ambient measurement dimension. For this work we
assume that the dynamics of the problem under consideration is evolving on a
low-dimensional subspace that is well characterized by a fast decaying singular
value spectrum
Stabilizer states and Clifford operations for systems of arbitrary dimensions, and modular arithmetic
We describe generalizations of the Pauli group, the Clifford group and
stabilizer states for qudits in a Hilbert space of arbitrary dimension d. We
examine a link with modular arithmetic, which yields an efficient way of
representing the Pauli group and the Clifford group with matrices over the
integers modulo d. We further show how a Clifford operation can be efficiently
decomposed into one and two-qudit operations. We also focus in detail on
standard basis expansions of stabilizer states.Comment: 10 pages, RevTe
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