Computational Optimization and Applications

, Volume 63, Issue 2, pp 333–364 | Cite as

Eigenvalue, quadratic programming, and semidefinite programming relaxations for a cut minimization problem

  • Ting Kei Pong
  • Hao Sun
  • Ningchuan Wang
  • Henry Wolkowicz


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.


Vertex separators Eigenvalue bounds Semidefinite programming bounds Graph partitioning Large scale 

Mathematics Subject Classification

05C70 15A42 90C22 90C27 90C59 



T. K. Pong was supported partly by a research grant from Hong Kong Polytechnic University. He was also supported as a PIMS postdoctoral fellow at Department of Computer Science, University of British Columbia, Vancouver, during the early stage of the preparation of the manuscript. Research of H. Sun supported by an Undergraduate Student Research Award from The Natural Sciences and Engineering Research Council of Canada. Research of N. Wang supported by The Natural Sciences and Engineering Research Council of Canada and by the U.S. Air Force Office of Scientific Research. H. Wolkowincz: Research supported in part by The Natural Sciences and Engineering Research Council of Canada and by the U.S. Air Force Office of Scientific Research.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Ting Kei Pong
    • 1
  • Hao Sun
    • 2
  • Ningchuan Wang
    • 2
  • Henry Wolkowicz
    • 2
  1. 1.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHung HomHong Kong
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

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