22,872 research outputs found
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
On a decomposition of regular domains into John domains with uniform constants
We derive a decomposition result for regular, two-dimensional domains into
John domains with uniform constants. We prove that for every simply connected
domain with -boundary there is a corresponding
partition with such
that each component is a John domain with a John constant only depending on
. The result implies that many inequalities in Sobolev spaces such as
Poincar\'e's or Korn's inequality hold on the partition of for uniform
constants, which are independent of
Shorter tours and longer detours: Uniform covers and a bit beyond
Motivated by the well known four-thirds conjecture for the traveling salesman
problem (TSP), we study the problem of {\em uniform covers}. A graph
has an -uniform cover for TSP (2EC, respectively) if the everywhere
vector (i.e. ) dominates a convex combination of
incidence vectors of tours (2-edge-connected spanning multigraphs,
respectively). The polyhedral analysis of Christofides' algorithm directly
implies that a 3-edge-connected, cubic graph has a 1-uniform cover for TSP.
Seb\H{o} asked if such graphs have -uniform covers for TSP for
some . Indeed, the four-thirds conjecture implies that such
graphs have 8/9-uniform covers. We show that these graphs have 18/19-uniform
covers for TSP. We also study uniform covers for 2EC and show that the
everywhere 15/17 vector can be efficiently written as a convex combination of
2-edge-connected spanning multigraphs.
For a weighted, 3-edge-connected, cubic graph, our results show that if the
everywhere 2/3 vector is an optimal solution for the subtour linear programming
relaxation, then a tour with weight at most 27/19 times that of an optimal tour
can be found efficiently. Node-weighted, 3-edge-connected, cubic graphs fall
into this category. In this special case, we can apply our tools to obtain an
even better approximation guarantee.
To extend our approach to input graphs that are 2-edge-connected, we present
a procedure to decompose an optimal solution for the subtour relaxation for TSP
into spanning, connected multigraphs that cover each 2-edge cut an even number
of times. Using this decomposition, we obtain a 17/12-approximation algorithm
for minimum weight 2-edge-connected spanning subgraphs on subcubic,
node-weighted graphs
Convex Polytopes and Quasilattices from the Symplectic Viewpoint
We construct, for each convex polytope, possibly nonrational and nonsimple, a
family of compact spaces that are stratified by quasifolds, i.e. each of these
spaces is a collection of quasifolds glued together in an suitable way. A
quasifold is a space locally modelled on modulo the action of a
discrete, possibly infinite, group. The way strata are glued to each other also
involves the action of an (infinite) discrete group. Each stratified space is
endowed with a symplectic structure and a moment mapping having the property
that its image gives the original polytope back. These spaces may be viewed as
a natural generalization of symplectic toric varieties to the nonrational
setting.Comment: LaTeX, 29 pages. Revised version: TITLE changed, reorganization of
notations and exposition, added remarks and reference
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