Before revision:  For WECP we develop a bi-kernel with 4^k vertices, and an algorithm with runtime 2^{ O (k^{3/2}w^{1/2}\log(k/w))}n^{O(1)}, where w is the maximum edge weight.
After revision:  For WECP we develop a bi-kernel with 4^k vertices, and an algorithm with runtime 2^{ \mathcal{O (k^{3/2}w^{1/2}\log(k/w))}n^{O(1)}, where w is the maximum edge weight.