4,524 research outputs found
Computing with Tangles
Tangles of graphs have been introduced by Robertson and Seymour in the
context of their graph minor theory. Tangles may be viewed as describing
"k-connected components" of a graph (though in a twisted way). They play an
important role in graph minor theory. An interesting aspect of tangles is that
they cannot only be defined for graphs, but more generally for arbitrary
connectivity functions (that is, integer-valued submodular and symmetric set
functions).
However, tangles are difficult to deal with algorithmically. To start with,
it is unclear how to represent them, because they are families of separations
and as such may be exponentially large. Our first contribution is a data
structure for representing and accessing all tangles of a graph up to some
fixed order.
Using this data structure, we can prove an algorithmic version of a very
general structure theorem due to Carmesin, Diestel, Harman and Hundertmark (for
graphs) and Hundertmark (for arbitrary connectivity functions) that yields a
canonical tree decomposition whose parts correspond to the maximal tangles.
(This may be viewed as a generalisation of the decomposition of a graph into
its 3-connected components.
Hypergraphs with Edge-Dependent Vertex Weights: p-Laplacians and Spectral Clustering
We study p-Laplacians and spectral clustering for a recently proposed
hypergraph model that incorporates edge-dependent vertex weights (EDVW). These
weights can reflect different importance of vertices within a hyperedge, thus
conferring the hypergraph model higher expressivity and flexibility. By
constructing submodular EDVW-based splitting functions, we convert hypergraphs
with EDVW into submodular hypergraphs for which the spectral theory is better
developed. In this way, existing concepts and theorems such as p-Laplacians and
Cheeger inequalities proposed under the submodular hypergraph setting can be
directly extended to hypergraphs with EDVW. For submodular hypergraphs with
EDVW-based splitting functions, we propose an efficient algorithm to compute
the eigenvector associated with the second smallest eigenvalue of the
hypergraph 1-Laplacian. We then utilize this eigenvector to cluster the
vertices, achieving higher clustering accuracy than traditional spectral
clustering based on the 2-Laplacian. More broadly, the proposed algorithm works
for all submodular hypergraphs that are graph reducible. Numerical experiments
using real-world data demonstrate the effectiveness of combining spectral
clustering based on the 1-Laplacian and EDVW
Discrete Convex Functions on Graphs and Their Algorithmic Applications
The present article is an exposition of a theory of discrete convex functions
on certain graph structures, developed by the author in recent years. This
theory is a spin-off of discrete convex analysis by Murota, and is motivated by
combinatorial dualities in multiflow problems and the complexity classification
of facility location problems on graphs. We outline the theory and algorithmic
applications in combinatorial optimization problems
On the Convergence Rate of Decomposable Submodular Function Minimization
Submodular functions describe a variety of discrete problems in machine
learning, signal processing, and computer vision. However, minimizing
submodular functions poses a number of algorithmic challenges. Recent work
introduced an easy-to-use, parallelizable algorithm for minimizing submodular
functions that decompose as the sum of "simple" submodular functions.
Empirically, this algorithm performs extremely well, but no theoretical
analysis was given. In this paper, we show that the algorithm converges
linearly, and we provide upper and lower bounds on the rate of convergence. Our
proof relies on the geometry of submodular polyhedra and draws on results from
spectral graph theory.Comment: 17 pages, 3 figure
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