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The "exact subgraph" approach was recently introduced as a hierarchicalscheme to get increasingly tight semidefinite programming relaxations ofseveral NP-hard graph optimization problems. Solving these relaxations is acomputational challenge because of the potentially large number of violatedsubgraph constraints. We introduce a computational framework for theserelaxations designed to cope with these difficulties. We suggest a partialLagrangian dual, and exploit the fact that its evaluation decomposes intoseveral independent subproblems. This opens the way to use the bundle methodfrom non-smooth optimization to minimize the dual function. Finallycomputational experiments on the Max-Cut, stable set and coloring problem showthe excellent quality of the bounds obtained with this approach.

A Computational Study of Exact Subgraph Based SDP Bounds for Max-Cut, Stable Set and Coloring