6,740 research outputs found
FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension
We show the existence of a fully polynomial-time approximation scheme (FPTAS)
for the problem of maximizing a non-negative polynomial over mixed-integer sets
in convex polytopes, when the number of variables is fixed. Moreover, using a
weaker notion of approximation, we show the existence of a fully
polynomial-time approximation scheme for the problem of maximizing or
minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes,
when the number of variables is fixed.Comment: 16 pages, 4 figures; to appear in Mathematical Programmin
Asymptotic properties of entanglement polytopes for large number of qubits
Entanglement polytopes have been recently proposed as the way of witnessing
the SLOCC multipartite entanglement classes using single particle information.
We present first asymptotic results concerning feasibility of this approach for
large number of qubits. In particular we show that entanglement polytopes of
-qubit system accumulate in the distance from the
point corresponding to the maximally mixed reduced one-qubit density matrices.
This implies existence of a possibly large region where many entanglement
polytopes overlap, i.e where the witnessing power of entanglement polytopes is
weak. Moreover, the witnessing power cannot be strengthened by any entanglement
distillation protocol as for large the required purity is above current
capability.Comment: 5 pages, 4 figure
A dice probability problem
Two different approaches to a probability problem involving convex polytopes lead to a geometric proof of an integral geometric result about mixed surface areas. The proof can be modified to cover the corresponding results about mixed volume
Random Walks and Mixed Volumes of Hypersimplices
Below is a method for relating a mixed volume computation for polytopes
sharing many facet directions to a symmetric random walk. The example of
permutahedra and particularly hypersimplices is expanded upon.Comment: 6 page
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