I am trying to understand the connection between The W-hierarchy as presented in chapter 13 of this book by Cygan et al. and the notion of the NP problems.
Is the existence of an FPT algorithm for a problem in W suggests that P=NP? Why?
For example, assuming I have an FPT algorithm for the k-Clique problem. Can I prove the P=NP? The algorithm run time will still be exponential. Only now, it will depend on $k$.