1,182 research outputs found
Decomposing Cubic Graphs into Connected Subgraphs of Size Three
Let be the set of connected graphs of size 3. We
study the problem of partitioning the edge set of a graph into graphs taken
from any non-empty . The problem is known to be NP-complete for
any possible choice of in general graphs. In this paper, we assume that
the input graph is cubic, and study the computational complexity of the problem
of partitioning its edge set for any choice of . We identify all polynomial
and NP-complete problems in that setting, and give graph-theoretic
characterisations of -decomposable cubic graphs in some cases.Comment: to appear in the proceedings of COCOON 201
Graph kernels between point clouds
Point clouds are sets of points in two or three dimensions. Most kernel
methods for learning on sets of points have not yet dealt with the specific
geometrical invariances and practical constraints associated with point clouds
in computer vision and graphics. In this paper, we present extensions of graph
kernels for point clouds, which allow to use kernel methods for such ob jects
as shapes, line drawings, or any three-dimensional point clouds. In order to
design rich and numerically efficient kernels with as few free parameters as
possible, we use kernels between covariance matrices and their factorizations
on graphical models. We derive polynomial time dynamic programming recursions
and present applications to recognition of handwritten digits and Chinese
characters from few training examples
On the Approximability of Digraph Ordering
Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute
a labeling maximizing the number of forward edges, i.e.
edges (u,v) such that (u) < (v). For different values of k, this
reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work
studies the approximability of Max-k-Ordering and its generalizations,
motivated by their applications to job scheduling with soft precedence
constraints. We give an LP rounding based 2-approximation algorithm for
Max-k-Ordering for any k={2,..., n}, improving on the known
2k/(k-1)-approximation obtained via random assignment. The tightness of this
rounding is shown by proving that for any k={2,..., n} and constant
, Max-k-Ordering has an LP integrality gap of 2 -
for rounds of the
Sherali-Adams hierarchy.
A further generalization of Max-k-Ordering is the restricted maximum acyclic
subgraph problem or RMAS, where each vertex v has a finite set of allowable
labels . We prove an LP rounding based
approximation for it, improving on the
approximation recently given by Grandoni et al.
(Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact,
our approximation algorithm also works for a general version where the
objective counts the edges which go forward by at least a positive offset
specific to each edge.
The minimization formulation of digraph ordering is DAG edge deletion or
DED(k), which requires deleting the minimum number of edges from an n-vertex
directed acyclic graph (DAG) to remove all paths of length k. We show that
both, the LP relaxation and a local ratio approach for DED(k) yield
k-approximation for any .Comment: 21 pages, Conference version to appear in ESA 201
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