3,380 research outputs found
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
Entanglement Witnesses in Spin Models
We construct entanglement witnesses using fundamental quantum operators of
spin models which contain two-particle interactions and posses a certain
symmetry. By choosing the Hamiltonian as such an operator, our method can be
used for detecting entanglement by energy measurement. We apply this method to
the cubic Heisenberg lattice model in a magnetic field, the XY model and other
familiar spin systems. Our method is used to obtain a temperature bound for
separable states for systems in thermal equilibrium. We also study the
Bose-Hubbard model and relate its energy minimum for separable states to the
minimum obtained from the Gutzwiller ansatz.Comment: 5 pages including 3 figures, revtex4; some typos correcte
Entanglement Detection in the Stabilizer Formalism
We investigate how stabilizer theory can be used for constructing sufficient
conditions for entanglement. First, we show how entanglement witnesses can be
derived for a given state, provided some stabilizing operators of the state are
known. These witnesses require only a small effort for an experimental
implementation and are robust against noise. Second, we demonstrate that also
nonlinear criteria based on uncertainty relations can be derived from
stabilizing operators. These criteria can sometimes improve the witnesses by
adding nonlinear correction terms. All our criteria detect states close to
Greenberger-Horne-Zeilinger states, cluster and graph states. We show that
similar ideas can be used to derive entanglement conditions for states which do
not fit the stabilizer formalism, such as the three-qubit W state. We also
discuss connections between the witnesses and some Bell inequalities.Comment: 15 pages including 2 figures, revtex4; typos corrected, presentation
improved; to appear in PR
New bounds on the average distance from the Fermat-Weber center of a planar convex body
The Fermat-Weber center of a planar body is a point in the plane from
which the average distance to the points in is minimal. We first show that
for any convex body in the plane, the average distance from the
Fermat-Weber center of to the points of is larger than , where is the diameter of . This proves a conjecture
of Carmi, Har-Peled and Katz. From the other direction, we prove that the same
average distance is at most . The new bound substantially improves the previous bound of
due to
Abu-Affash and Katz, and brings us closer to the conjectured value of . We also confirm the upper bound conjecture for centrally
symmetric planar convex bodies.Comment: 13 pages, 2 figures. An earlier version (now obsolete): A. Dumitrescu
and Cs. D. T\'oth: New bounds on the average distance from the Fermat-Weber
center of a planar convex body, in Proceedings of the 20th International
Symposium on Algorithms and Computation (ISAAC 2009), 2009, LNCS 5878,
Springer, pp. 132-14
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