Add Catgory (assign) to Post Programmatically (without editing)

I am looking for a way to assign (add) a category to a list of posts without editing each and every post – even by using bulk edit.

Why?
Because doing it manually (even using bulk) will update the published time and I do not want that.

Any idea as to how this can be done? I understand that I need an array of post ID:s, which I have. But how do I do this using my theme’s functions file?