4,137 research outputs found
On Generalizations of Network Design Problems with Degree Bounds
Iterative rounding and relaxation have arguably become the method of choice
in dealing with unconstrained and constrained network design problems. In this
paper we extend the scope of the iterative relaxation method in two directions:
(1) by handling more complex degree constraints in the minimum spanning tree
problem (namely, laminar crossing spanning tree), and (2) by incorporating
`degree bounds' in other combinatorial optimization problems such as matroid
intersection and lattice polyhedra. We give new or improved approximation
algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure
Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces
Polyhedra and their arrangements have intrigued humankind since the ancient
Greeks and are today important motifs in condensed matter, with application to
many classes of liquids and solids. Yet, little is known about the
thermodynamically stable phases of polyhedrally-shaped building blocks, such as
faceted nanoparticles and colloids. Although hard particles are known to
organize due to entropy alone, and some unusual phases are reported in the
literature, the role of entropic forces in connection with polyhedral shape is
not well understood. Here, we study thermodynamic self-assembly of a family of
truncated tetrahedra and report several atomic crystal isostructures, including
diamond, {\beta}-tin, and high- pressure lithium, as the polyhedron shape
varies from tetrahedral to octahedral. We compare our findings with the densest
packings of the truncated tetrahedron family obtained by numerical compression
and report a new space filling polyhedron, which has been overlooked in
previous searches. Interestingly, the self-assembled structures differ from the
densest packings. We show that the self-assembled crystal structures can be
understood as a tendency for polyhedra to maximize face-to-face alignment,
which can be generalized as directional entropic forces.Comment: Article + supplementary information. 23 pages, 10 figures, 2 table
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
On the equivalence between the cell-based smoothed finite element method and the virtual element method
We revisit the cell-based smoothed finite element method (SFEM) for
quadrilateral elements and extend it to arbitrary polygons and polyhedrons in
2D and 3D, respectively. We highlight the similarity between the SFEM and the
virtual element method (VEM). Based on the VEM, we propose a new stabilization
approach to the SFEM when applied to arbitrary polygons and polyhedrons. The
accuracy and the convergence properties of the SFEM are studied with a few
benchmark problems in 2D and 3D linear elasticity. Later, the SFEM is combined
with the scaled boundary finite element method to problems involving
singularity within the framework of the linear elastic fracture mechanics in
2D
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