The claims you're making in this subthread are not precise enough to be correct. If you only allow a constant finite amount of time to solve a given problem, th

The claims you're making in this subthread are not precise enough to be correct. If you only allow a constant finite amount of time to solve a given problem, then aysmptotics guarantee nothing. An O(1) time could solve fewer instances than an O(2^2^n) time algorithm. It is also not true that the asymptotically faster algorithm "can solve more instances as the allocated time increases." These statements are only true if you add the phrase "sufficiently large" to them. So asymptotically faster algorithms can solve more instances as the allocated time bound increases, provided that the time bound is sufficiently large. But how large that "sufficiently large" needs to be is arbitrary, so you can't say that Babai's algorithm is practical without either precisely studying the constants involved in the algorithm (which Babai did to some extent, but omitted in the notation for clarity), or doing empirical analysis, which is unlikely to ever happen with Babai's algorithm.