Strengthening Chvátal-Gomory Cuts for the Stable Set Problem

The stable set problem is a well-known \(\mathcal{NP}\)-hard combinatorial optimization problem. As well as being hard to solve (or even approximate) in theory, it is often hard to solve in practice. The main difficulty is that upper bounds based on linear programming (LP) tend to be weak, whereas upper bounds based on semidefinite programming (SDP) take a long time to compute. We propose a new method to strengthen the LP-based upper bounds. The key idea is to take violated Chvatal-Gomory cuts and then strengthen their right-hand sides. Although the strengthening problem is itself \(\mathcal{NP}\)-hard, it can be solved reasonably quickly in practice. As a result, the overall procedure proves to be capable of yielding competitive upper bounds in reasonable computing times.

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