Local algorithms for independent sets are half-optimal

We show that the largest density of factor of i.i.d. independent sets in the dd-regular tree is asymptotically at most (logd)/d(log⁡d)/d as d→∞d→∞. This matches the lower bound given by previous constructions. It follows that the largest independent sets given by local algorithms on random dd-regular graphs have the same asymptotic density. In contrast, the density of the largest independent sets in these graphs is asymptotically 2(logd)/d2(log⁡d)/d. We prove analogous results for Poisson–Galton–Watson trees, which yield bounds for local algorithms on sparse Erdős–Renyi graphs.