A Bundle Approach for SDPs with Exact Subgraph Constraints

Abstract

The ‘exact subgraph’ approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational challenge because of the potentially large number of violated subgraph constraints. We introduce a computational framework for these relaxations designed to cope with these difficulties. We suggest a partial Lagrangian dual, and exploit the fact that its evaluation decomposes into two independent subproblems. This opens the way to use the bundle method from non-smooth optimization to minimize the dual function. Computational experiments on the Max-Cut, stable set and coloring problem show the efficiency of this approach.

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 764759 and the Austrian Science Fund (FWF): I 3199-N31 and P 28008-N35. We thank three anonymous referees for their constructive comments which substantially helped to improve the presentation of our material.

A Tables

Table 1. The running times for one Max-Cut and one stable set instance with different fixed sets of ESCs. The graphs of order $n=100$ are from the Erdős-Rényi model.

Table 2. The deviation of the ESB to $z_{mc}$ for several Max-Cut instances.

Table 3. The gap of the ESB to $z_{mc}$ for two Max-Cut instances.

Table 4. Tighten $\vartheta (G)$ towards $\alpha (G)$ for several instances for 10 cycles.

Table 5. Maximum found projection distance of $X_{I}$ to ${{\,\mathrm{STAB}\,}}^{2}(G_{I})$ for the computations of Table 4.

Table 6. Tighten $\vartheta (G)$ towards $\chi (G)$ for several instances for 10 cycles.

Table 7. Maximum found projection distance of $X_{I}$ to ${{\,\mathrm{COL}\,}}(G_{I})$ for the computations of Table 6.