Why do professors ask candidates at PhD interviews what specific problems they are interested in? I am currently interviewing in several universities across the U.S. for PhD programs in computer science. In every one of my interviews so far, there are always 1 or 2 professors that are eager to know which specific problems I am interested in researching (and it normally comes out as the most important question they are going to ask me). This question stumps me every time because I am not trying to enter a PhD program and start working on a specific problem right away. Instead, I want to do some lab rotations first and get a deeper feeling for the work of the institution's faculty (that works on my subarea of interest) and then, after these rotations, pick a specific problem (perhaps this is the wrong motivation?).

It does not seem to me that interest is a reliable determiner of success in completing a PhD. I could say that I am interested in Dr.X's research on complex manifolds because I skimmed through some of his papers. However, from my personal experience, I think there is a very big discrepancy between having read a little bit about a topic and claiming it as an interest, and actually doing research in that area. In other words, one might be interested in doing research in modelling the response of cancer patients to drug Y (because they want to defeat cancer or something similar) but a lot of times that person might realize that the actual research is far from his expectations and end up switching to different area. 

I could list a few other reasons why I think interest is not a good measure of student's commitment but I might be wrong since professors like to ask this question. Why is this and what would be an honest way to answer it?
Other than the previous answer, we ask this question because a candidate may be interested in an area that is not in fact well represented in the department. If someone wanted to do research on numerical methods for optimization, my department has nobody to advise such a thesis -- but the candidate may not be aware of this.

I personally ask this question because too many graduate students come to me and ask whether I could be their adviser and then expect me to pose a problem they should work on for the 3-4 years. But I don't want to do that: if a student doesn't know what they want to do for research, yes, sure, I can give them a topic, but if it doesn't match their interest, it's going to be a painful 3-4 years for everyone involved. So I typically want that my students come up with topics they find interesting themselves, and then we can see how that fits into my research group and program.