Optimal Low-Degree Hardness of Maximum Independent Set

10/13/2020

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by   Alexander S. Wein, et al.

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We study the algorithmic task of finding a large independent set in a sparse Erdős-Rényi random graph with n vertices and average degree d. The maximum independent set is known to have size (2 log d / d)n in the double limit n →∞ followed by d →∞, but the best known polynomial-time algorithms can only find an independent set of half-optimal size (log d / d)n. We show that the class of low-degree polynomial algorithms can find independent sets of half-optimal size but no larger, improving upon a result of Gamarnik, Jagannath, and the author. This generalizes earlier work by Rahman and Virág, which proved the analogous result for the weaker class of local algorithms.