A Continuous Quadratic Programming Formulation of the Vertex Separator Problem

The Vertex Separator Problem (VSP) for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets of roughly equal size. In a recent paper (Optimality Conditions For Maximizing a Function Over a Polyhedron, Mathematical Programming, 2013, doi: 10.1007/s10107-013-0644-1), the authors announced a new continuous bilinear quadratic programming formulation of the VSP, and they used this quadratic programming problem to illustrate the new optimality conditions. The current paper develops conditions for the equivalence between this continuous quadratic program and the vertex separator problem, and it examines the relationship between the continuous formulation of the VSP and continuous quadratic programming formulations for both the edge separator problem and maximum clique problem.