1,292 research outputs found
Colorful Strips
Given a planar point set and an integer , we wish to color the points with
colors so that any axis-aligned strip containing enough points contains all
colors. The goal is to bound the necessary size of such a strip, as a function
of . We show that if the strip size is at least , such a coloring
can always be found. We prove that the size of the strip is also bounded in any
fixed number of dimensions. In contrast to the planar case, we show that
deciding whether a 3D point set can be 2-colored so that any strip containing
at least three points contains both colors is NP-complete.
We also consider the problem of coloring a given set of axis-aligned strips,
so that any sufficiently covered point in the plane is covered by colors.
We show that in dimensions the required coverage is at most .
Lower bounds are given for the two problems. This complements recent
impossibility results on decomposition of strip coverings with arbitrary
orientations. Finally, we study a variant where strips are replaced by wedges
Fixed parameter tractable algorithms in combinatorial topology
To enumerate 3-manifold triangulations with a given property, one typically
begins with a set of potential face pairing graphs (also known as dual
1-skeletons), and then attempts to flesh each graph out into full
triangulations using an exponential-time enumeration. However, asymptotically
most graphs do not result in any 3-manifold triangulation, which leads to
significant "wasted time" in topological enumeration algorithms. Here we give a
new algorithm to determine whether a given face pairing graph supports any
3-manifold triangulation, and show this to be fixed parameter tractable in the
treewidth of the graph.
We extend this result to a "meta-theorem" by defining a broad class of
properties of triangulations, each with a corresponding fixed parameter
tractable existence algorithm. We explicitly implement this algorithm in the
most generic setting, and we identify heuristics that in practice are seen to
mitigate the large constants that so often occur in parameterised complexity,
highlighting the practicality of our techniques.Comment: 16 pages, 9 figure
The number of crossings in multigraphs with no empty lens
Let be a multigraph with vertices and edges, drawn in the
plane such that any two parallel edges form a simple closed curve with at least
one vertex in its interior and at least one vertex in its exterior. Pach and
T\'oth (2018) extended the Crossing Lemma of Ajtai et al. (1982) and Leighton
(1983) by showing that if no two adjacent edges cross and every pair of
nonadjacent edges cross at most once, then the number of edge crossings in
is at least , for a suitable constant . The situation
turns out to be quite different if nonparallel edges are allowed to cross any
number of times. It is proved that in this case the number of crossings in
is at least . The order of magnitude of this bound
cannot be improved.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
A graph interpretation of the least squares ranking method
The paper aims at analyzing the least squares ranking method for generalized
tournaments with possible missing and multiple paired comparisons. The
bilateral relationships may reflect the outcomes of a sport competition,
product comparisons, or evaluation of political candidates and policies. It is
shown that the rating vector can be obtained as a limit point of an iterative
process based on the scores in almost all cases. The calculation is interpreted
on an undirected graph with loops attached to some nodes, revealing that the
procedure takes into account not only the given object's results but also the
strength of objects compared with it. We explore the connection between this
method and another procedure defined for ranking the nodes in a digraph, the
positional power measure. The decomposition of the least squares solution
offers a number of ways to modify the method
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