Fractional Langevin Dynamics for Combinatorial Optimization via Polynomial-Time Escape

Langevin Dynamics (LD) and its discrete proposal have been widely applied in the field of Combinatorial Optimization (CO). Both sampling-based and data-driven approaches have benefited significantly from these methods. However, LD's reliance on Gaussian noise limits its ability to escape narrow local optima, requires costly parallel chains, and performs poorly in rugged landscapes or with non-strict constraints. These challenges have impeded the development of more advanced approaches. To address these issues, we introduce Fractional Langevin Dynamics (FLD) for CO, replacing Gaussian noise with $\alpha$-stable L\'evy noise. FLD can escape from local optima more readily via L\'evy flights, and in multiple-peak CO problems with high potential barriers it exhibits a polynomial escape time that outperforms the exponential escape time of LD. Moreover, FLD coincides with LD when $\alpha = 2$, and by tuning $\alpha$ it can be adapted to a wider range of complex scenarios in the CO fields. We provide theoretical proof that our method offers enhanced exploration capabilities and improved convergence. Experimental results on the Maximum Independent Set, Maximum Clique, and Maximum Cut problems demonstrate that incorporating FLD advances both sampling-based and data-driven approaches, achieving state-of-the-art (SOTA) performance in most of the experiments.