A Multi-Level Algorithm For Partitioning Graphs
08 Dec 1995-pp 28-28
TL;DR: A multilevel algorithm for graph partitioning in which the graph is approximated by a sequence of increasingly smaller graphs, and the smallest graph is then partitioned using a spectral method, and this partition is propagated back through the hierarchy of graphs.
Abstract: The graph partitioning problem is that of dividing the vertices of a graph into sets of specified sizes such that few edges cross between sets. This NP-complete problem arises in many important scientific and engineering problems. Prominent examples include the decomposition of data structures for parallel computation, the placement of circuit elements and the ordering of sparse matrix computations. We present a multilevel algorithm for graph partitioning in which the graph is approximated by a sequence of increasingly smaller graphs. The smallest graph is then partitioned using a spectral method, and this partition is propagated back through the hierarchy of graphs. A variant of the Kernighan-Lin algorithm is applied periodically to refine the partition. The entire algorithm can be implemented to execute in time proportional to the size of the original graph. Experiments indicate that, relative to other advanced methods, the multilevel algorithm produces high quality partitions at low cost.
Citations
More filters
TL;DR: This work presents a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of theSize of the final partition obtained after multilevel refinement, and presents a much faster variation of the Kernighan--Lin (KL) algorithm for refining during uncoarsening.
Abstract: Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc. of the 6th SIAM Conference on Parallel Processing for Scientific Computing, 1993, 445--452; Hendrickson and Leland, A Multilevel Algorithm for Partitioning Graphs, Tech. report SAND 93-1301, Sandia National Laboratories, Albuquerque, NM, 1993]. From the early work it was clear that multilevel techniques held great promise; however, it was not known if they can be made to consistently produce high quality partitions for graphs arising in a wide range of application domains. We investigate the effectiveness of many different choices for all three phases: coarsening, partition of the coarsest graph, and refinement. In particular, we present a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of the size of the final partition obtained after multilevel refinement. We also present a much faster variation of the Kernighan--Lin (KL) algorithm for refining during uncoarsening. We test our scheme on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that our scheme produces partitions that are consistently better than those produced by spectral partitioning schemes in substantially smaller time. Also, when our scheme is used to compute fill-reducing orderings for sparse matrices, it produces orderings that have substantially smaller fill than the widely used multiple minimum degree algorithm.
5,629 citations
Cites background or methods from "A Multi-Level Algorithm For Partiti..."
...These are called multilevel graph partitioning schemes [4, 7, 19, 20, 26, 10, 43]....
[...]
...We also present a new variation of the Kernighan-Lin algorithm for refining the partition during the uncoarsening phase that is much faster than the Kernighan-Lin refinement used in [26]....
[...]
...In this paper we build on the work of Hendrickson and Leland....
[...]
...Since both the coarsening phase and the refinement phase with the Kernighan-Lin heuristic take roughly the same amount of time, the overall run-time of the multilevel scheme of [26] cannot be reduced significantly....
[...]
...Compared with the multilevel scheme of [26], our scheme is about two to seven times faster, and is consistently better in terms of cut size....
[...]
TL;DR: This paper presents and study a class of graph partitioning algorithms that reduces the size of the graph by collapsing vertices and edges, they find ak-way partitioning of the smaller graph, and then they uncoarsen and refine it to construct ak- way partitioning for the original graph.
1,715 citations
TL;DR: This paper employs approximation algorithms for the graph-partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities, and defines the network community profile plot, which characterizes the "best" possible community—according to the conductance measure—over a wide range of size scales.
Abstract: A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. Rather than defining a procedure to extract sets of nodes from a graph and then attempting to interpret these sets as "real" communities, we employ approximation algorithms for the graph-partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be i...
1,660 citations
Cites background from "A Multi-Level Algorithm For Partiti..."
...mbinatorial quantity; and it has a very natural interpretation in terms of random walks on the interaction graph. Moreover, since there exist a rich suite of both theoretical and practical algorithms [87, 149, 107, 108, 17, 95, 96, 162, 54], we can for point (4) compare and contrast several methods to approximately optimize it. To illustrate conductance, note that of the three 5-node sets A, B, and C illustrated in the graph in Figure 1...
[...]
Posted Content•
TL;DR: In this article, the authors employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities.
Abstract: A large body of work has been devoted to defining and identifying clusters or communities in social and information networks. We explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. We employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the "best" possible community--according to the conductance measure--over a wide range of size scales. We study over 100 large real-world social and information networks. Our results suggest a significantly more refined picture of community structure in large networks than has been appreciated previously. In particular, we observe tight communities that are barely connected to the rest of the network at very small size scales; and communities of larger size scales gradually "blend into" the expander-like core of the network and thus become less "community-like." This behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, it is exactly the opposite of what one would expect based on intuition from expander graphs, low-dimensional or manifold-like graphs, and from small social networks that have served as testbeds of community detection algorithms. We have found that a generative graph model, in which new edges are added via an iterative "forest fire" burning process, is able to produce graphs exhibiting a network community profile plot similar to what we observe in our network datasets.
1,555 citations
TL;DR: This paper develops a fast high-quality multilevel algorithm that directly optimizes various weighted graph clustering objectives, such as the popular ratio cut, normalized cut, and ratio association criteria, and demonstrates that the algorithm is applicable to large-scale clustering tasks such as image segmentation, social network analysis, and gene network analysis.
Abstract: A variety of clustering algorithms have recently been proposed to handle data that is not linearly separable; spectral clustering and kernel k-means are two of the main methods In this paper, we discuss an equivalence between the objective functions used in these seemingly different methods - in particular, a general weighted kernel k-means objective is mathematically equivalent to a weighted graph clustering objective We exploit this equivalence to develop a fast high-quality multilevel algorithm that directly optimizes various weighted graph clustering objectives, such as the popular ratio cut, normalized cut, and ratio association criteria This eliminates the need for any eigenvector computation for graph clustering problems, which can be prohibitive for very large graphs Previous multilevel graph partitioning methods such as Metis have suffered from the restriction of equal-sized clusters; our multilevel algorithm removes this restriction by using kernel k-means to optimize weighted graph cuts Experimental results show that our multilevel algorithm outperforms a state-of-the-art spectral clustering algorithm in terms of speed, memory usage, and quality We demonstrate that our algorithm is applicable to large-scale clustering tasks such as image segmentation, social network analysis, and gene network analysis
1,038 citations
Additional excerpts
...Ç...
[...]
References
More filters
TL;DR: A heuristic method for partitioning arbitrary graphs which is both effective in finding optimal partitions, and fast enough to be practical in solving large problems is presented.
Abstract: We consider the problem of partitioning the nodes of a graph with costs on its edges into subsets of given sizes so as to minimize the sum of the costs on all edges cut. This problem arises in several physical situations — for example, in assigning the components of electronic circuits to circuit boards to minimize the number of connections between boards. This paper presents a heuristic method for partitioning arbitrary graphs which is both effective in finding optimal partitions, and fast enough to be practical in solving large problems.
5,082 citations
01 Jan 1982
TL;DR: An iterative mincut heuristic for partitioning networks is presented whose worst case computation time, per pass, grows linearly with the size of the network.
Abstract: An iterative mincut heuristic for partitioning networks is presented whose worst case computation time, per pass, grows linearly with the size of the network. In practice, only a very small number of passes are typically needed, leading to a fast approximation algorithm for mincut partitioning. To deal with cells of various sizes, the algorithm progresses by moving one cell at a time between the blocks of the partition while maintaining a desired balance based on the size of the blocks rather than the number of cells per block. Efficient data structures are used to avoid unnecessary searching for the best cell to move and to minimize unnecessary updating of cells affected by each move.
2,463 citations
TL;DR: This paper shows that a number of NP - complete problems remain NP -complete even when their domains are substantially restricted, and determines essentially the lowest possible upper bounds on node degree for which the problems remainNP -complete.
2,200 citations
TL;DR: In this paper, it is shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph.
Abstract: The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is, shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorith...
1,762 citations
TL;DR: Numerical comparisons on large-scale two- and three-dimensional problems demonstrate the superiority of the new spectral bisection algorithm.
834 citations
"A Multi-Level Algorithm For Partiti..." refers background or methods in this paper
...We use a spectral method [13, 21] for this problem, but in principle any partitioning algorithm could be used....
[...]
...More detailed descriptions of spectral bisection can be found in [10, 11, 13, 19, 21]....
[...]
...The rst competing algorithm is inertial bisection, descriptions of which can be found in [21, 27]....
[...]