15,965 research outputs found
QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails
Rods are less fragile than spheres: Structural relaxation in dense liquids composed of anisotropic particles
We perform extensive molecular dynamics simulations of dense liquids composed
of bidisperse dimer- and ellipse-shaped particles in 2D that interact via
repulsive contact forces. We measure the structural relaxation times obtained
from the long-time decay of the self-part of the intermediate scattering
function for the translational and rotational degrees of freedom (DOF) as a
function of packing fraction \phi, temperature T, and aspect ratio \alpha. We
are able to collapse the \phi and T-dependent structural relaxation times for
disks, and dimers and ellipses over a wide range of \alpha, onto a universal
scaling function {\cal F}_{\pm}(|\phi-\phi_0|,T,\alpha), which is similar to
that employed in previous studies of dense liquids composed of purely repulsive
spherical particles in 3D. {\cal F_{\pm}} for both the translational and
rotational DOF are characterized by the \alpha-dependent scaling exponents \mu
and \delta and packing fraction \phi_0(\alpha) that signals the crossover in
the scaling form {\cal F}_{\pm} from hard-particle dynamics to super-Arrhenius
behavior for each aspect ratio. We find that the fragility at \phi_0,
m(\phi_0), decreases monotonically with increasing aspect ratio for both
ellipses and dimers. Moreover, the results for the slow dynamics of dense
liquids composed of dimer- and ellipse-shaped particles are qualitatively the
same, despite the fact that zero-temperature static packings of dimers are
isostatic, while static packings of ellipses are hypostatic.Comment: 10 pages, 17 figures, and 1 tabl
- …