3,306 research outputs found
A survey on algorithmic aspects of modular decomposition
The modular decomposition is a technique that applies but is not restricted
to graphs. The notion of module naturally appears in the proofs of many graph
theoretical theorems. Computing the modular decomposition tree is an important
preprocessing step to solve a large number of combinatorial optimization
problems. Since the first polynomial time algorithm in the early 70's, the
algorithmic of the modular decomposition has known an important development.
This paper survey the ideas and techniques that arose from this line of
research
Graph-Based Classification of Omnidirectional Images
Omnidirectional cameras are widely used in such areas as robotics and virtual
reality as they provide a wide field of view. Their images are often processed
with classical methods, which might unfortunately lead to non-optimal solutions
as these methods are designed for planar images that have different geometrical
properties than omnidirectional ones. In this paper we study image
classification task by taking into account the specific geometry of
omnidirectional cameras with graph-based representations. In particular, we
extend deep learning architectures to data on graphs; we propose a principled
way of graph construction such that convolutional filters respond similarly for
the same pattern on different positions of the image regardless of lens
distortions. Our experiments show that the proposed method outperforms current
techniques for the omnidirectional image classification problem
Arboricity, h-Index, and Dynamic Algorithms
In this paper we present a modification of a technique by Chiba and Nishizeki
[Chiba and Nishizeki: Arboricity and Subgraph Listing Algorithms, SIAM J.
Comput. 14(1), pp. 210--223 (1985)]. Based on it, we design a data structure
suitable for dynamic graph algorithms. We employ the data structure to
formulate new algorithms for several problems, including counting subgraphs of
four vertices, recognition of diamond-free graphs, cop-win graphs and strongly
chordal graphs, among others. We improve the time complexity for graphs with
low arboricity or h-index.Comment: 19 pages, no figure
Fully polynomial FPT algorithms for some classes of bounded clique-width graphs
Parameterized complexity theory has enabled a refined classification of the
difficulty of NP-hard optimization problems on graphs with respect to key
structural properties, and so to a better understanding of their true
difficulties. More recently, hardness results for problems in P were achieved
using reasonable complexity theoretic assumptions such as: Strong Exponential
Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to
these assumptions, many graph theoretic problems do not admit truly
subquadratic algorithms, nor even truly subcubic algorithms (Williams and
Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central
technique used to tackle the difficulty of the above mentioned problems is
fixed-parameter algorithms for polynomial-time problems with polynomial
dependency in the fixed parameter (P-FPT). This technique was introduced by
Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016)
and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this
technique to clique-width, another important graph parameter, remained to be
done. In this paper we study several graph theoretic problems for which
hardness results exist such as cycle problems (triangle detection, triangle
counting, girth, diameter), distance problems (diameter, eccentricities, Gromov
hyperbolicity, betweenness centrality) and maximum matching. We provide
hardness results and fully polynomial FPT algorithms, using clique-width and
some of its upper-bounds as parameters (split-width, modular-width and
-sparseness). We believe that our most important result is an -time algorithm for computing a maximum matching where
is either the modular-width or the -sparseness. The latter generalizes
many algorithms that have been introduced so far for specific subclasses such
as cographs, -lite graphs, -extendible graphs and -tidy
graphs. Our algorithms are based on preprocessing methods using modular
decomposition, split decomposition and primeval decomposition. Thus they can
also be generalized to some graph classes with unbounded clique-width
Exploring Subexponential Parameterized Complexity of Completion Problems
Let be a family of graphs. In the -Completion problem,
we are given a graph and an integer as input, and asked whether at most
edges can be added to so that the resulting graph does not contain a
graph from as an induced subgraph. It appeared recently that special
cases of -Completion, the problem of completing into a chordal graph
known as Minimum Fill-in, corresponding to the case of , and the problem of completing into a split graph,
i.e., the case of , are solvable in parameterized
subexponential time . The exploration of this
phenomenon is the main motivation for our research on -Completion.
In this paper we prove that completions into several well studied classes of
graphs without long induced cycles also admit parameterized subexponential time
algorithms by showing that:
- The problem Trivially Perfect Completion is solvable in parameterized
subexponential time , that is -Completion for , a cycle and a path on four
vertices.
- The problems known in the literature as Pseudosplit Completion, the case
where , and Threshold Completion, where , are also solvable in time .
We complement our algorithms for -Completion with the following
lower bounds:
- For , , , and
, -Completion cannot be solved in time
unless the Exponential Time Hypothesis (ETH) fails.
Our upper and lower bounds provide a complete picture of the subexponential
parameterized complexity of -Completion problems for .Comment: 32 pages, 16 figures, A preliminary version of this paper appeared in
the proceedings of STACS'1
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