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Title: Eigenvalue, quadratic programming, and semidefinite programming relaxations for a cut minimization problem
Authors: Pong, TK 
Sun, H
Wang, N
Wolkowicz, H
Issue Date: Mar-2016
Source: Computational optimization and applications, Mar. 2016, v. 63, no. 2, p. 333-364
Abstract: We consider the problem of partitioning the node set of a graph into k sets of given sizes in order to minimize the cut obtained using (removing) the kth set. If the resulting cut has value 0, then we have obtained a vertex separator. This problem is closely related to the graph partitioning problem. In fact, the model we use is the same as that for the graph partitioning problem except for a different quadratic objective function. We look at known and new bounds obtained from various relaxations for this NP-hard problem. This includes: the standard eigenvalue bound, projected eigenvalue bounds using both the adjacency matrix and the Laplacian, quadratic programming (QP) bounds based on recent successful QP bounds for the quadratic assignment problems, and semidefinite programming bounds. We include numerical tests for large and huge problems that illustrate the efficiency of the bounds in terms of strength and time.
Keywords: Eigenvalue bounds
Graph partitioning
Large scale
Semidefinite programming bounds
Vertex separators
Publisher: Springer
Journal: Computational optimization and applications 
ISSN: 0926-6003
EISSN: 1573-2894
DOI: 10.1007/s10589-015-9779-8
Rights: © Springer Science+Business Media New York 2015
This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use (, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at:
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