Continuous Formulations for Combinatorial Optimization Problems Based on the Probabilistic Method

We present a unified framework of exact continuous formulations for a wide range of classical graph optimization problems, including the minimum (double) dominating set, vertex cover, maximum independent set, clique, s-plex, and max-cut, by leveraging the probabilistic method, and introduce several novel weighted extensions. The approach yields multilinear polynomial programs over the unit hypercube whose optimal values coincide with the corresponding combinatorial invariants. To assess practical performance, we conduct numerical experiments using a modern global solver (Gurobi) and five state-of-the-art local solvers (CONOPT, IPOPT, KNITRO, LOQO, SNOPT). We find that high-degree multilinear objectives can severely impede convergence, but that systematic degree reduction via Boole’s inequality drastically improves solution quality and robustness. The obtained results are promising and encourage a more detailed investigation of continuous formulations based on the probabilistic method in the context of solving discrete optimization problems.