448 research outputs found
Relaxation-Based Coarsening for Multilevel Hypergraph Partitioning
Multilevel partitioning methods that are inspired by principles of
multiscaling are the most powerful practical hypergraph partitioning solvers.
Hypergraph partitioning has many applications in disciplines ranging from
scientific computing to data science. In this paper we introduce the concept of
algebraic distance on hypergraphs and demonstrate its use as an algorithmic
component in the coarsening stage of multilevel hypergraph partitioning
solvers. The algebraic distance is a vertex distance measure that extends
hyperedge weights for capturing the local connectivity of vertices which is
critical for hypergraph coarsening schemes. The practical effectiveness of the
proposed measure and corresponding coarsening scheme is demonstrated through
extensive computational experiments on a diverse set of problems. Finally, we
propose a benchmark of hypergraph partitioning problems to compare the quality
of other solvers
Hypergraph Modelling for Geometric Model Fitting
In this paper, we propose a novel hypergraph based method (called HF) to fit
and segment multi-structural data. The proposed HF formulates the geometric
model fitting problem as a hypergraph partition problem based on a novel
hypergraph model. In the hypergraph model, vertices represent data points and
hyperedges denote model hypotheses. The hypergraph, with large and
"data-determined" degrees of hyperedges, can express the complex relationships
between model hypotheses and data points. In addition, we develop a robust
hypergraph partition algorithm to detect sub-hypergraphs for model fitting. HF
can effectively and efficiently estimate the number of, and the parameters of,
model instances in multi-structural data heavily corrupted with outliers
simultaneously. Experimental results show the advantages of the proposed method
over previous methods on both synthetic data and real images.Comment: Pattern Recognition, 201
Efficient Flow-based Approximation Algorithms for Submodular Hypergraph Partitioning via a Generalized Cut-Matching Game
In the past 20 years, increasing complexity in real world data has lead to
the study of higher-order data models based on partitioning hypergraphs.
However, hypergraph partitioning admits multiple formulations as hyperedges can
be cut in multiple ways. Building upon a class of hypergraph partitioning
problems introduced by Li & Milenkovic, we study the problem of minimizing
ratio-cut objectives over hypergraphs given by a new class of cut functions,
monotone submodular cut functions (mscf's), which captures hypergraph expansion
and conductance as special cases.
We first define the ratio-cut improvement problem, a family of local
relaxations of the minimum ratio-cut problem. This problem is a natural
extension of the Andersen & Lang cut improvement problem to the hypergraph
setting. We demonstrate the existence of efficient algorithms for approximately
solving this problem. These algorithms run in almost-linear time for the case
of hypergraph expansion, and when the hypergraph rank is at most .
Next, we provide an efficient -approximation algorithm for finding
the minimum ratio-cut of . We generalize the cut-matching game framework of
Khandekar et. al. to allow for the cut player to play unbalanced cuts, and
matching player to route approximate single-commodity flows. Using this
framework, we bootstrap our algorithms for the ratio-cut improvement problem to
obtain approximation algorithms for minimum ratio-cut problem for all mscf's.
This also yields the first almost-linear time -approximation
algorithms for hypergraph expansion, and constant hypergraph rank.
Finally, we extend a result of Louis & Makarychev to a broader set of
objective functions by giving a polynomial time -approximation algorithm for the minimum ratio-cut problem based on
rounding -metric embeddings.Comment: Comments and feedback welcom
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
- …