135 research outputs found
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
Parameterization and approximation are two popular ways of coping with
NP-hard problems. More recently, the two have also been combined to derive many
interesting results. We survey developments in the area both from the
algorithmic and hardness perspectives, with emphasis on new techniques and
potential future research directions
Treewidth reduction for constrained separation and bipartization problems
We present a method for reducing the treewidth of a graph while preserving
all the minimal separators. This technique turns out to be very useful
for establishing the fixed-parameter tractability of constrained separation and
bipartization problems. To demonstrate the power of this technique, we prove
the fixed-parameter tractability of a number of well-known separation and
bipartization problems with various additional restrictions (e.g., the vertices
being removed from the graph form an independent set). These results answer a
number of open questions in the area of parameterized complexity.Comment: STACS final version of our result. For the complete description of
the result please see version
Open Problems in (Hyper)Graph Decomposition
Large networks are useful in a wide range of applications. Sometimes problem
instances are composed of billions of entities. Decomposing and analyzing these
structures helps us gain new insights about our surroundings. Even if the final
application concerns a different problem (such as traversal, finding paths,
trees, and flows), decomposing large graphs is often an important subproblem
for complexity reduction or parallelization. This report is a summary of
discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph
Decomposition" and presents currently open problems and future directions in
the area of (hyper)graph decomposition
Graph Learning and Its Applications: A Holistic Survey
Graph learning is a prevalent domain that endeavors to learn the intricate
relationships among nodes and the topological structure of graphs. These
relationships endow graphs with uniqueness compared to conventional tabular
data, as nodes rely on non-Euclidean space and encompass rich information to
exploit. Over the years, graph learning has transcended from graph theory to
graph data mining. With the advent of representation learning, it has attained
remarkable performance in diverse scenarios, including text, image, chemistry,
and biology. Owing to its extensive application prospects, graph learning
attracts copious attention from the academic community. Despite numerous works
proposed to tackle different problems in graph learning, there is a demand to
survey previous valuable works. While some researchers have perceived this
phenomenon and accomplished impressive surveys on graph learning, they failed
to connect related objectives, methods, and applications in a more coherent
way. As a result, they did not encompass current ample scenarios and
challenging problems due to the rapid expansion of graph learning. Different
from previous surveys on graph learning, we provide a holistic review that
analyzes current works from the perspective of graph structure, and discusses
the latest applications, trends, and challenges in graph learning.
Specifically, we commence by proposing a taxonomy from the perspective of the
composition of graph data and then summarize the methods employed in graph
learning. We then provide a detailed elucidation of mainstream applications.
Finally, based on the current trend of techniques, we propose future
directions.Comment: 20 pages, 7 figures, 3 table
Ollivier-Ricci Curvature for Hypergraphs: A Unified Framework
Bridging geometry and topology, curvature is a powerful and expressiveinvariant. While the utility of curvature has been theoretically andempirically confirmed in the context of manifolds and graphs, itsgeneralization to the emerging domain of hypergraphs has remained largelyunexplored. On graphs, Ollivier-Ricci curvature measures differences betweenrandom walks via Wasserstein distances, thus grounding a geometric concept inideas from probability and optimal transport. We develop ORCHID, a flexibleframework generalizing Ollivier-Ricci curvature to hypergraphs, and prove thatthe resulting curvatures have favorable theoretical properties. Throughextensive experiments on synthetic and real-world hypergraphs from differentdomains, we demonstrate that ORCHID curvatures are both scalable and useful toperform a variety of hypergraph tasks in practice.<br
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