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1995, Annals of Operations Research
Let G = (N; E) be a given undirected graph. We present several new techniques for partitioning the node set N into k disjoint subsets of speci ed sizes. These techniques involve eigenvalue bounds and tools from continuous optimization. Comparisons with examples taken from the literature show these techniques to be very successful.
Mathematical Programming, 1994
Let G = (N; E) be an edge-weighted undirected graph. The graph partitioning problem is the problem of partitioning the node set N into k disjoint subsets of speci ed sizes so as to minimize the total weight of the edges connecting nodes in distinct subsets of the partition. We present a numerical study on the use of eigenvalue-based techniques to nd upper and lower bounds for this problem. Results for bisecting graphs with up to several thousand nodes are given, and for small graphs some trisection results are presented. We show that the techniques are very robust and consistently produce upper and lower bounds having a relative gap of typically a few percentage points.
Discrete Applied Mathematics
Given an undirected edge weighted graph, the graph partitioning problem consists in determining a partition of the node set of the graph into subsets of prescribed sizes, so as to maximize the sum of the weights of the edges having both endpoints in the same subset. We introduce a new class of bounds for this problem relying on the full spectral information of the weighted adjacency matrix A. The expression of these bounds involves the eigenvalues and particular geometrical parameters defined using the eigenvectors of A. A connection is established between these parameters and the maximum cut problem. We report computational results showing that the new bounds compare favorably with previous bounds in the literature.
Mathematical Programming, 2014
The graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the sum of weights of edges joining different sets is optimized. In this paper we simplify a known matrixlifting semidefinite programming relaxation of the graph partition problem for several classes of graphs and also show how to aggregate additional triangle and independent set constraints for graphs with symmetry. We present an eigenvalue bound for the graph partition problem of a strongly regular graph, extending a similar result for the equipartition problem. We also derive a linear programming bound of the graph partition problem for certain Johnson and Kneser graphs. Using what we call the Laplacian algebra of a graph, we derive an eigenvalue bound for the graph partition problem that is the first known closed form bound that is applicable to any graph, thereby extending a well-known result in spectral graph theory. Finally, we strengthen a known semidefinite programming relaxation of a specific quadratic assignment problem and the above-mentioned matrix-lifting semidefinite programming relaxation by adding two constraints that correspond to assigning two vertices of the graph to different parts of the partition. This strengthening performs well on highly symmetric graphs when other relaxations provide weak or trivial bounds.
Information Processing Letters, 1992
arXiv (Cornell University), 2023
The problem of multiway partitioning of an undirected graph is considered. A spectral method is used, where the k > 2 largest eigenvalues of the normalized adjacency matrix (equivalently, the k smallest eigenvalues of the normalized graph Laplacian) are computed. It is shown that the information necessary for partitioning is contained in the subspace spanned by the k eigenvectors. The partitioning is encoded in a matrix Ψ in indicator form, which is computed by approximating the eigenvector matrix by a product of Ψ and an orthogonal matrix. A measure of the distance of a graph to being k-partitionable is defined, as well as two cut (cost) functions, for which Cheeger inequalities are proved; thus the relation between the eigenvalue and partitioning problems is established. Numerical examples are given that demonstrate that the partitioning algorithm is efficient and robust.
Proceedings of the International Conference on …, 1995
A survey of published methods for partitioning sparse arrays is presented. These include early attempts to describe the partitioning properties of eigenvectors of the adjacency matrix. More direct methods of partitioning are developed by introducing the Laplacian of the adjacency matrix via the directed (signed) edge-vertex incidence matrix. It is shown that the Laplacian solves the minimization of total length of connections between adjacent nodes, which induces clustering of connected nodes by partitioning the underlying graph. Another matrix derived from the adjacency matrix is also introduced via the unsigned edge-vertex matrix. This (the Normal) matrix is not symmetric, and it also is shown to solve the minimization of total length in its own non-Euclidean metric. In this case partitions are induced by clustering the connected nodes. The Normal matrix is closely related to Correspondence Analysis.
2021
Partitioning a connected graph into k vertex-disjoint connected subgraphs of similar (or given) orders is a classical problem that has been intensively investigated since late seventies. Given a connected graph G = (V,E) and a weight function w : V → Q≥, a connected k-partition of G is a partition of V such that each class induces a connected subgraph. The balanced connected k-partition problem consists in finding a connected k-partition in which every class has roughly the same weight. To model this concept of balance, one may seek connected k-partitions that either maximize the weight of a lightest class (max-min BCPk) or minimize the weight of a heaviest class (min-max BCPk). Such problems are equivalent when k = 2, but they are different when k ≥ 3. In this work, we propose a simple pseudo-polynomial k 2 -approximation algorithm for min-max BCPk which runs in time O(W |V ||E|), where W = ∑ v∈V w(v). Based on this algorithm and using a scaling technique, we design a (polynomial) ...
SIAM J. MATRIX ANAL. …, 1990
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 to 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
Journal of Graph Algorithms and Applications, 2000
This paper considers problems of the following type: given an edgeweighted k-colored input graph with maximum color class size c, find a minimum or maximum c-way cut such that each color class is totally partitioned. Equivalently, given a weighted complete k-partite graph, cover its vertices with a minimum number of disjoint cliques in such a way that the total weight of the cliques is maximized or minimized. Our study was motivated by some work called the index domain alignment problem [6], which shows its relevance to optimization of distributed computation. Solutions of these problems also have applications in logistics [3] and manufacturing systems [10]. In this paper, we design some approximation algorithms by extending the matching algorithms to these problems. Both theoretical and experimental results show that the algorithms we designed produce good approximations. Communicated by D. Eppstein.
Lecture Notes in Computer Science, 1999
Let an edge cut partition the vertex set of a graph into k disjoint subsets A1, . . . , A k with ||Ai| − |Aj|| ≤ 1. We consider the problem of determining the minimal size of such a cut for a given graph. For this we introduce a new lower bound method which is based on the solution of an extremal set problem and present bounds for some graph classes based on Hamming graphs.
arXiv (Cornell University), 2022
The graph partition problem (GPP) aims at clustering the vertex set of a graph into a fixed number of disjoint subsets of given sizes such that the sum of weights of edges joining different sets is minimized. This paper investigates the quality of doubly nonnegative (DNN) relaxations, i.e., relaxations having matrix variables that are both positive semidefinite and nonnegative, strengthened by additional polyhedral cuts for two variations of the GPP: the k-equipartition and the graph bisection problem. After reducing the size of the relaxations by facial reduction, we solve them by a cutting-plane algorithm that combines an augmented Lagrangian method with Dykstra's projection algorithm. Since many components of our algorithm are general, the algorithm is suitable for solving various DNN relaxations with a large number of cutting planes. We are the first to show the power of DNN relaxations with additional cutting planes for the GPP on large benchmark instances up to 1,024 vertices. Computational results show impressive improvements in strengthened DNN bounds.
2004
This report describes a graph partitioning algorithm based on spectral factorization that can be implemented very efficiently with just a hand full of MATLAB commands. The algorithm is closely related to the one proposed by Phillips and Kokotović [4] for state-aggregation in Markov chains. The appendix contains a MATLAB script that implements the algorithm. This algorithm is available online at [3].
2015
In real life, there are many problems like shortest path, graph coloring, travelling Salesmen problem (TSP) etc, thus providing solution to each problem is nearly or highly impossible with the help of traditional methods in reasonable amount of time. But it may be possible with the help of heuristic approach. It provides solution but don’t guarantee optimal solution. Graph partitioning problems are NP-Complete problems, partitioning graph into p-partitions using multilevel method, spectral method etc. for various purposes .Here we are studying several techniques to partition graph.
Mathematical Programming, 1998
In this paper we consider the problem of k-partitioning the nodes of a graph with capacity restrictions on the sum of the node weights in each subset of the partition, and the objective of minimizing the sum of the costs of the edges between the subsets of the partition. Based on a study of valid inequalities, we present a variety of separation heuristics for cycle, cycle with ears, knapsack tree and path-block-cycle inequalities among others. The separation heuristics, plus primal heuristics, have been implemented in a branch-andcut routine using a formulation including variables for the edges with nonzero costs and node partition variables. Results are presented for three classes of problems: equipartitioning problems arising in nite element methods and partitioning problems associated with electronic circuit layout and compiler design.
We survey recent trends in practical algorithms for balanced graph partitioning, point to applications and discuss future research directions.
2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence), 2008
In this paper, a new meta-method based on the physical nuclear process is presented. This meta-method called Fusion-Fission is applied to the two different class of graph partitioning problems. This paper presents results found by this method in comparison with results of classical methods for an air traffic management problem, an image segmentation problem and applied to classical benchmarks. All of these applications of the Fusion-Fission method are successful and the results found by this method outperform state-of-the-art graph partitioning packages both on classical benchmarks and on the air traffic management problem.
SIAM Journal on Optimization, 2007
We consider 3-partitioning the vertices of a graph into sets S 1 , S 2 and S 3 of specified cardinalities, such that the total weight of all edges joining S 1 and S 2 is minimized. This problem is closely related to several NP-hard problems like determining the bandwidth or finding a vertex separator in a graph. We show that this problem can be formulated as a linear program over the cone of completely positive matrices, leading in a natural way to semidefinite relaxations of the problem. We show in particular that the spectral relaxation introduced by Helmberg et al. (1995) can equivalently be formulated as a semidefinite program. Finally we propose a tightened version of this semidefinite program and show on some small instances that this new bound is a significant improvement over the spectral bound.
INFORMS Journal on Computing, 2014
We derive a new semidefinite programming relaxation for the general graph partition problem (GPP). Our relaxation is based on matrix lifting with matrix variable having order equal to the number of vertices of the graph. We show that this relaxation is equivalent to the Frieze-Jerrum relaxation [A. Frieze and M. Jerrum. Improved approximation algorithms for max k-cut and max bisection. Algorithmica, 18(1):67-81, 1997] for the maximum k-cut problem with an additional constraint that involves the restrictions on the subset sizes. Since the new relaxation does not depend on the number of subsets k into which the graph should be partitioned we are able to compute bounds for large k. We compare theoretically and numerically the new relaxation with other SDP relaxations for the GPP. The results show that our relaxation provides competitive bounds and is solved significantly faster than any other known SDP bound for the general GPP.
Journal of Global Optimization, 2010
The graph partitioning problem is to partition the vertex set of a graph into a number of nonempty subsets so that the total weight of edges connecting distinct subsets is minimized. Previous research requires the input of cardinalities of subsets or the number of subsets for equipartition. In this paper, the problem is formulated as a zero-one quadratic programming problem without the input of cardinalities. We also present three equivalent zero-one linear integer programming reformulations. Because of its importance in data biclustering, the bipartite graph partitioning is also studied. Several new methods to determine the number of subsets and the cardinalities are presented for practical applications. In addition, hierarchy partitioning and partitioning of bipartite graphs without reordering one vertex set, are studied.
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