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We consider the problem of partitioning the node set of a graph into $k$ setsof given sizes in order to \emph{minimize the cut} obtained using (removing)the $k$-th set. If the resulting cut has value $0$, then we have obtained avertex separator. This problem is closely related to the graph partitioningproblem. In fact, the model we use is the same as that for the graphpartitioning problem except for a different \emph{quadratic} objectivefunction. We look at known and new bounds obtained from various relaxations forthis NP-hard problem. This includes: the standard eigenvalue bound, projectedeigenvalue bounds using both the adjacency matrix and the Laplacian, quadraticprogramming (QP) bounds based on recent successful QP bounds for the quadraticassignment problems, and semidefinite programming bounds. We include numericaltests for large and \emph{huge} problems that illustrate the efficiency of thebounds in terms of strength and time.

Eigenvalue, Quadratic Programming, and Semidefinite Programming Bounds for a Cut Minimization Problem