Is an improved algorithm for a non-hot problem publishable? I have been working on a research problem independent of any advisor for the past few months and I think I have a result. The problem I am investigating  has n and K as input. The naive approach would take O(n!K+2), which is ridiculously bad. I managed to come up with, and proof the correctness of a 2-approximation algorithm that runs in O(Kn2).

However, this is absolutely not a hot topic. Researchers in it are not a lot. On the other hand, there are practical applications of my algorithm (off the top of my head, I can think of three significant ones). I am also positive that my algorithm beats all published approximation algorithms as well.

So my question is: Would a reputable journal accept my paper given that it isn’t really a hot topic now?
Journals often publish articles that show something that is "interesting" or "elegant" or "unexpected", even if the topic may not necessarily be "hot". (Of course, it is easier to publish something that combines any of these attributes with "hot", but not all articles are on "hot" topics.) So, I think there is hope in your case and I would give it a try.

I would, however, suggest that you do a careful literature review. You compare your algorithm with a "naive" one, but that may not mean that the "naive" algorithm is what people would use or what is considered state of the art. For example, the traveling salesman problem is known to be an exponentially complicated problem if you are looking for an exact solution, but that doesn't mean that that is what people use when they actually want to solve such problems -- very good approximate solutions can be found quite efficiently.

In other words, in your paper you will not only have to explain your new method, but you should point out that people have actually been using the "ridiculously bad" method. Otherwise, you will hear from the reviewers that you make your approach sound artificially good by comparing it with something no reasonable person would use.