We give a deterministic quasi-polynomial time algorithm for approximating the largest independent set in a graph. We show that if an $N$-vertex graph contains an independent set of size $2^{-t} N$, then we can find an independent set of size $2^t$ in quasi-polynomial time, for all $t$ up to a constant factor of $(\log N)^{1/3}$. This gives an approximation factor of $N / \exp((\log N)^{1/3})$, which has the same shape as known hardness results for quasi-polynomial time algorithms, up to a different exponent of the logarithm.

The algorithm builds on a structural principle: a graph containing a large independent set must also contain a large sparse structured set, and such a set can be found efficiently. A greedy algorithm then extracts an independent set from it. Our proof draws on tools in additive combinatorics, in particular spreadness and sifting, to obtain the combinatorial structural results. We then convert these to efficient algorithms using pseudo-marginals obtained by semidefinite programming.