Min-cut clustering

Johnson, Ellis J. L.; Mehrotra, Anuj; Nemhauser, George L.

doi:10.1007/bf01585164

Cited by 191 publications

(87 citation statements)

References 8 publications

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“…in 0-1 variables a j [93,82,68]. For the minimum sum-of-squares problem, it reduces to a hyperbolic 0-1 program, in view of Huyghens' theorem, which states that the sum of squared distances to the centroid is equal to the sum of squared distances between entities divided by the cardinality of the cluster:…”

Section: Column Generation Methodsmentioning

confidence: 99%

“…This rule appears to be more efficient than the previous one [67] and variants of it have been applied with success in several papers on scheduling problems, e.g., [33]. Nevertheless, some recent column generation methods for clustering,e.g., [93,82] still stopped after solution of the master's problem relaxation or used some heuristic from that point. In a recent survey [4], the name "branch-and-price" has been proposed for combination of column generation and branch-and-bound.…”

Section: Column Generation Methodsmentioning

confidence: 99%

“…Cutting-planes were also used in [82] to solve, in moderate time, the auxiliary problem in a column generation approach (see next subsection) to a constrained minimum-sum-of-cuts partitioning problem (called min-cut clustering).…”

Section: Cutting Planesmentioning

confidence: 99%

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Cluster analysis and mathematical programming

Hansen

Jaumard

1997

Mathematical Programming

413

251

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Les textes publiés dans la série des rapports de recherche HEC n'engagent que la responsabilité de leurs auteurs. La publication de ces rapports de recherche bénéficie d'une subvention du Fonds F.C.A.R. Cluster Analysis and Mathematical ProgrammingPierre Hansen GERAD andÉcole des HautesÉtudes Commerciales Montréal, CanadaBrigitte Jaumard GERAD andÉcole Polytechnique de Montréal Canada February, 1997Les Cahiers du GERAD G-97-10Abstract Given a set of entities, Cluster Analysis aims at finding subsets, called clusters, which are homogeneous and/or well separated. As many types of clustering and criteria for homogeneity or separation are of interest, this is a vast field. A survey is given from a mathematical programming viewpoint.Steps of a clustering study, types of clustering and criteria are discussed. Then algorithms for hierarchical, partitioning, sequential, and additive clustering are studied. Emphasis is on solution methods, i.e., dynamic programming, graph theoretical algorithms, branch-and-bound, cutting planes, column generation and heuristics.

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Section: Column Generation Methodsmentioning

confidence: 99%

Section: Column Generation Methodsmentioning

confidence: 99%

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Cluster analysis and mathematical programming

Hansen

Jaumard

1997

Mathematical Programming

413

251

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“…Exponential size RMP linear programs are polynomially solvable by column generation under the assumption that the pricing problem is solvable in polynomial time (Mehlhorn and Ziegelmann, 2000;Minoux, 1987). Conversely, solving an RMP is N Phard, if the pricing problem is (Johnson, Mehrotra, and Nemhauser, 1993).…”

Section: A Dual Point Of Viewmentioning

confidence: 99%

Selected Topics in Column Generation

Lübbecke

Desrosiers

2005

Operations Research

848

488

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Dantzig-Wolfe decomposition and column generation, devised for linear programs, is a success story in large scale integer programming. We outline and relate the approaches, and survey mainly recent contributions, not yet found in textbooks. We emphasize the growing understanding of the dual point of view, which has brought considerable progress to the column generation theory and practice. It stimulated careful initializations, sophisticated solution techniques for the restricted master problem and subproblem, as well as better overall performance. Thus, the dual perspective is an ever recurring concept in our "selected topics."

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“…The same approach is followed by L. Brunetta, M. Conforti, and G. Rinaldi in [2]. In [15], E. Johnson, A. Mehrotra, and G. Nemhauser elaborate on a column generation approach for the graph partitioning problem.…”

Section: Introductionmentioning

confidence: 99%

Multicommodity Flow Approximation Used for Exact Graph Partitioning

Sellmann

Sensen

Timajev

2003

Algorithms - ESA 2003

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Abstract. We present a fully polynomial-time approximation scheme for a multicommodity flow problem that yields lower bounds of the graph bisection problem. We compare the approximation algorithm with Lagrangian relaxation based cost-decomposition approaches and linear programming software when embedded in an exact branch&bound approach for graph bisection. It is shown that the approximation algorithm is clearly superior in this context. Furthermore, we present a new practical addition to the approximation algorithm which improves its performance distinctly. Finally, we prove the performance of the graph bisection algorithm using multicommodity flow approximation by computing formerly unknown bisection widths of some DeBruijn-and Shuffle-Exchange-Graphs.

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Min-cut clustering

Cited by 191 publications

References 8 publications

Cluster analysis and mathematical programming

Cluster analysis and mathematical programming

Selected Topics in Column Generation

Multicommodity Flow Approximation Used for Exact Graph Partitioning

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