On the Optimality of the Median Cut Spectral Bisection Graph Partitioning Method

  title={On the Optimality of the Median Cut Spectral Bisection Graph Partitioning Method},
  author={Tony F. Chan and Patrick Ciarlet and W. K. Szeto},
  journal={SIAM J. Sci. Comput.},
Recursive spectral bisection (RSB) is a heuristic technique for finding a minimum cut graph bisection. To use this method the second eigenvector of the Laplacian of the graph is computed and from it a bisection is obtained. The most common method is to use the median of the components of the second eigenvector to induce a bisection. We prove here that this median cut method is optimal in the sense that the partition vector induced by it is the closest partition vector, in any ls norm, for $s… 
On the maximal error of spectral approximation of graph bisection
Spectral graph bisections are a popular heuristic aimed at approximating the solution of the NP-complete graph bisection problem. This technique, however, does not always provide a robust tool for
Embedded in the Shadow of the Separator
This work proves that, for any separator in the graph, at least one of the two separated node sets is embedded in the shadow (with the origin being the light source) of the convex hull of the separator.
Spectral bisection of graphs and connectedness
Abstract We present a refinement of the work of Miroslav Fiedler regarding bisections of irreducible matrices. We consider graph bisections as defined by the cut set consisting of characteristic
A Graph Based Method for Generating the Fiedler Vector of Irregular Problems
New algorithms for spectral graph partitioning are presented, which calculate the Fiedler vector of the original graph and use the information about the problem in the form of a preconditioner for the graph Laplacian.
A Novel Way of Computing Dissimilarities between Nodes of a Graph, with Application to Collaborative Filtering
A new perspective on characterizing the similarity between elements of a database or nodes of a weighted, undirected, graph based on a Markov-chain model of random walk through the database, which nicely fits into the so-called “statistical relational learning” framework.
Random-Walk Computation of Similarities between Nodes of a Graph with Application to Collaborative Recommendation
The model, which nicely fits into the so-called "statistical relational learning" framework, could also be used to compute document or word similarities, and could be applied to machine-learning and pattern-recognition tasks involving a relational database.
Nodal decompositions of graphs
Abstract A nodal domain of a function is a maximally connected subset of the domain for which the function does not change sign. Courant's nodal domain theorem gives a bound on the number of nodal
A novel way of computing similarities between nodes of a graph, with application to collaborative filtering and subspace projection of the graph nodes
This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted and undirected graph. It is based on a Markov-chain model
Distributed computation of the Fiedler vector with application to topology inference in ad hoc networks
This paper considers ad hoc networks where the nodes can process and exchange data in a synchronous fashion, and proposes a distributed algorithm for in-network estimation of the Fiedler vector and the algebraic connectivity of the corresponding network graph.
Edge partitioning of large graphs
The novel fashion vertex-cut is applied, instead of the traditional edge-cut method, for achieving balanced workload in distributed graph processing and the overhead of both communication and runtime can be decreased greatly, compared to existing approaches.


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, which can be used to compute good separators in grid graphs.
An Improved Spectral Graph Partitioning Algorithm for Mapping Parallel Computations
A new domain mapping algorithm is presented that extends recent work in which ideas from spectral graph theory have been applied to this problem and provides better decompositions arrived at more economically and robustly than with previous spectral methods.
The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Lapla- cian eigenvalue 2 and its relation to
An efficient heuristic procedure for partitioning graphs
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.
A projection technique for partitioning the nodes of a graph
Several new techniques for partitioning the node setN intok disjoint subsets of specified sizes are presented, which involve eigenvalue bounds and tools from continuous optimization.
Geometric algorithms for a minimum cost assignment problem
This work considers the minimum cost A-assignment problem, which is equivalent to the minimum weight one-to-many matching problem in a complete bipartite graph r = (A,B), where A and B have n and k nodes respectively, and gives an qkn + k“n”’)-tirne randomized algorithm, better than existing qkt$ + n2 log n)-time algorithm if ks nob.
Fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems
A multilevel version of RSB is introduced that attains about an order-of-magnitude improvement in run time on typical examples of unstructured meshes used in several large-scale scientific and engineering problems.
Performance of dynamic load balancing algorithms for unstructured mesh calculations
Three parallel algorithms, Orthogonal Recursive Bisection (ORB), Eigenvector Recursive bisection (ERB) and a simple parallelization of Simulated Annealing (SA) have been implemented for load balancing a dynamic unstructured triangular mesh on 16 processors of an NCUBE machine.
Partitioning of unstructured problems for parallel processing
Numerical comparisons on large-scale two- and three-dimensional problems demonstrate the superiority of the new spectral bisection algorithm.
An analysis of spectral graph partitioning via quadratic assignment problems,i n Domain Decomposition Methods in Science and Engineering
  • Proc. of the 7th Int'l Conf. on Domain Decomposition
  • 1995