Before revision:  The latter in particular improves the runtime for ECP to~2^{ O (k^{3/2}\log k)}n^{O(1)} . We also show that our algorithm necessarily needs a runtime of 2^{\Theta(k^{3/2\log k)}n^{O(1)} to solve ECP} .
After revision:  The latter in particular improves the runtime for ECP to~2^{ \mathcal{O (k^{3/2}\log k)}n^{O(1)} . We also show that our algorithm necessarily needs a runtime of 2^{\Theta(k^{3/2\log k)}n^{O(1)} to solve ECP} .