2,514 research outputs found
The Complexity of Cluster Vertex Splitting and Company
Clustering a graph when the clusters can overlap can be seen from three
different angles: We may look for cliques that cover the edges of the graph, we
may look to add or delete few edges to uncover the cluster structure, or we may
split vertices to separate the clusters from each other. Splitting a vertex
means to remove it and to add two new copies of and to make each previous
neighbor of adjacent with at least one of the copies. In this work, we
study the underlying computational problems regarding the three angles to
overlapping clusterings, in particular when the overlap is small. We show that
the above-mentioned covering problem, which also has been independently studied
in different contexts,is NP-complete. Based on a previous so-called
critical-clique lemma, we leverage our hardness result to show that Cluster
Editing with Vertex Splitting is also NP-complete, resolving an open question
by Abu-Khzam et al. [ISCO 2018]. We notice, however, that the proof of the
critical-clique lemma is flawed and we give a counterexample. Our hardness
result also holds under a version of the critical-clique lemma to which we
currently do not have a counterexample. On the positive side, we show that
Cluster Vertex Splitting admits a vertex-linear problem kernel with respect to
the number of splits.Comment: 30 pages, 9 figure
On the Threshold of Intractability
We study the computational complexity of the graph modification problems
Threshold Editing and Chain Editing, adding and deleting as few edges as
possible to transform the input into a threshold (or chain) graph. In this
article, we show that both problems are NP-complete, resolving a conjecture by
Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128,
2001). On the positive side, we show the problem admits a quadratic vertex
kernel. Furthermore, we give a subexponential time parameterized algorithm
solving Threshold Editing in time,
making it one of relatively few natural problems in this complexity class on
general graphs. These results are of broader interest to the field of social
network analysis, where recent work of Brandes (ISAAC, 2014) posits that the
minimum edit distance to a threshold graph gives a good measure of consistency
for node centralities. Finally, we show that all our positive results extend to
the related problem of Chain Editing, as well as the completion and deletion
variants of both problems
PACE solver description: KaPoCE: A heuristic cluster editing algorithm
The cluster editing problem is to transform an input graph into a cluster graph by performing a minimum number of edge editing operations. A cluster graph is a graph where each connected component is a clique. An edit operation can be either adding a new edge or removing an existing edge. In this write-up we outline the core techniques used in the heuristic cluster editing algorithm of the Karlsruhe and Potsdam Cluster Editing (KaPoCE) framework, submitted to the heuristic track of the 2021 PACE challenge
PACE Solver Description: KaPoCE: A Heuristic Cluster Editing Algorithm
The cluster editing problem is to transform an input graph into a cluster graph by performing a minimum number of edge editing operations. A cluster graph is a graph where each connected component is a clique. An edit operation can be either adding a new edge or removing an existing edge. In this write-up we outline the core techniques used in the heuristic cluster editing algorithm of the Karlsruhe and Potsdam Cluster Editing (KaPoCE) framework, submitted to the heuristic track of the 2021 PACE challenge
Linear Shape Deformation Models with Local Support Using Graph-based Structured Matrix Factorisation
Representing 3D shape deformations by linear models in high-dimensional space
has many applications in computer vision and medical imaging, such as
shape-based interpolation or segmentation. Commonly, using Principal Components
Analysis a low-dimensional (affine) subspace of the high-dimensional shape
space is determined. However, the resulting factors (the most dominant
eigenvectors of the covariance matrix) have global support, i.e. changing the
coefficient of a single factor deforms the entire shape. In this paper, a
method to obtain deformation factors with local support is presented. The
benefits of such models include better flexibility and interpretability as well
as the possibility of interactively deforming shapes locally. For that, based
on a well-grounded theoretical motivation, we formulate a matrix factorisation
problem employing sparsity and graph-based regularisation terms. We demonstrate
that for brain shapes our method outperforms the state of the art in local
support models with respect to generalisation ability and sparse shape
reconstruction, whereas for human body shapes our method gives more realistic
deformations.Comment: Please cite CVPR 2016 versio
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