We give an algorithm that takes as input a graph G with weights on the vertices and outputs a maximum weight independent set of G. If G does not contain any cycle on k or more vertices as an induced subgraph (G is a C ≥k -free graph), the algorithm runs in time n O(k 3 log 5 n) , and therefore for fixed k is a quasi-polynomial time algorithm. C ≥4 -free graphs (also known as chordal graphs) have a well known polynomial time algorithm. A subexponetial time algorithm for C ≥5 -free graphs (also known as long-hole-free graphs) was found in 2019 [Chudnovsky et al., Arxiv'19] followed by a polynomial time algorithm for C ≥5 -free graphs in 2020 [Abrishami et al., Arxiv'20]. For k > 5 only a quasi-polynomial time approximation scheme [Chudnovsky et al., SODA'20] was known. Our work is the first to exhibit conclusive evidence that Independent Set on C ≥k -free graphs is not NP-complete for any integer k. This also generalizes previous work of ours [Gartland and Lokshtanov, FOCS'20], with an additional factor of log 2 (n) in the exponent, where we provided a quasi-polynomial time algorithm for graphs that exclude a path on k vertices as an induced subgraph.

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