1,676 research outputs found
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
Beyond pairwise clustering
We consider the problem of clustering in domains where the affinity relations are not dyadic (pairwise), but rather triadic, tetradic or higher. The problem is an instance of the hypergraph partitioning problem. We propose a two-step algorithm for solving this problem. In the first step we use a novel scheme to approximate the hypergraph using a weighted graph. In the second step a spectral partitioning algorithm is used to partition the vertices of this graph. The algorithm is capable of handling hyperedges of all orders including order two, thus incorporating information of all orders simultaneously. We present a theoretical analysis that relates our algorithm to an existing hypergraph partitioning algorithm and explain the reasons for its superior performance. We report the performance of our algorithm on a variety of computer vision problems and compare it to several existing hypergraph partitioning algorithms
Approximation Algorithms for Hypergraph Small Set Expansion and Small Set Vertex Expansion
The expansion of a hypergraph, a natural extension of the notion of expansion
in graphs, is defined as the minimum over all cuts in the hypergraph of the
ratio of the number of the hyperedges cut to the size of the smaller side of
the cut. We study the Hypergraph Small Set Expansion problem, which, for a
parameter , asks to compute the cut having the least
expansion while having at most fraction of the vertices on the smaller
side of the cut. We present two algorithms. Our first algorithm gives an
approximation. The second algorithm finds
a set with expansion in a --uniform hypergraph with maximum degree
(where is the expansion of the optimal solution).
Using these results, we also obtain algorithms for the Small Set Vertex
Expansion problem: we get an
approximation algorithm and an algorithm that finds a set with vertex expansion
(where is the vertex expansion of the optimal
solution).
For , Hypergraph Small Set Expansion is equivalent to the
hypergraph expansion problem. In this case, our approximation factor of
for expansion in hypergraphs matches the corresponding
approximation factor for expansion in graphs due to ARV
Learning and Testing Variable Partitions
Let be a multivariate function from a product set to an
Abelian group . A -partition of with cost is a partition of
the set of variables into non-empty subsets such that is -close to
for some with
respect to a given error metric. We study algorithms for agnostically learning
partitions and testing -partitionability over various groups and error
metrics given query access to . In particular we show that
Given a function that has a -partition of cost , a partition
of cost can be learned in time
for any .
In contrast, for and learning a partition of cost is NP-hard.
When is real-valued and the error metric is the 2-norm, a
2-partition of cost can be learned in time
.
When is -valued and the error metric is Hamming
weight, -partitionability is testable with one-sided error and
non-adaptive queries. We also show that even
two-sided testers require queries when .
This work was motivated by reinforcement learning control tasks in which the
set of control variables can be partitioned. The partitioning reduces the task
into multiple lower-dimensional ones that are relatively easier to learn. Our
second algorithm empirically increases the scores attained over previous
heuristic partitioning methods applied in this context.Comment: Innovations in Theoretical Computer Science (ITCS) 202
Consistency of Spectral Hypergraph Partitioning under Planted Partition Model
Hypergraph partitioning lies at the heart of a number of problems in machine
learning and network sciences. Many algorithms for hypergraph partitioning have
been proposed that extend standard approaches for graph partitioning to the
case of hypergraphs. However, theoretical aspects of such methods have seldom
received attention in the literature as compared to the extensive studies on
the guarantees of graph partitioning. For instance, consistency results of
spectral graph partitioning under the stochastic block model are well known. In
this paper, we present a planted partition model for sparse random non-uniform
hypergraphs that generalizes the stochastic block model. We derive an error
bound for a spectral hypergraph partitioning algorithm under this model using
matrix concentration inequalities. To the best of our knowledge, this is the
first consistency result related to partitioning non-uniform hypergraphs.Comment: 35 pages, 2 figures, 1 tabl
Polyhedral Approaches to Hypergraph Partitioning and Cell Formation
Ankara : Department of Industrial Engineering and Institute of Engineering and Science, Bilkent University, 1994.Thesis (Ph.D.) -- -Bilkent University, 1994.Includes bibliographical references leaves 152-161Hypergraphs are generalizations of graphs in the sense that each hyperedge
can connect more than two vertices. Hypergraphs are used to describe manufacturing
environments and electrical circuits. Hypergraph partitioning in manufacturing
models cell formation in Cellular Manufacturing systems. Moreover,
hypergraph partitioning in VTSI design case is necessary to simplify the layout
problem. There are various heuristic techniques for obtaining non-optimal hypergraph
partitionings reported in the literature. In this dissertation research,
optimal seeking hypergraph partitioning approaches are attacked from polyhedral
combinatorics viewpoint.
There are two polytopes defined on r-uniform hypergraphs in which every
hyperedge has exactly r end points, in order to analyze partitioning related problems.
Their dimensions, valid inequality families, facet defining inequalities are
investigated, and experimented via random test problems.
Cell formation is the first stage in designing Cellular Manufacturing systems.
There are two new cell formation techniques based on combinatorial optimization
principles. One uses graph approximation, creation of a flow equivalent tree by
successively solving maximum flow problems and a search routine. The other
uses the polynomially solvable special case of the one of the previously discussed
polytopes. These new techniques are compared to six well-known cell formation
algorithms in terms of different efficiency measures according to randomly generated
problems. The results are analyzed statistically.Kandiller, LeventPh.D
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