When Is the Classroom Assignment Problem Hard?

The classroom assignment (or hotel room or interval scheduling) problem is to assign classes, which meet at different time intervals, to rooms. Two classes may not meet simultaneously in the same room, nor may a class meet in two different rooms. Thousands of colleges and secondary schools face this problem every semester. There has been some confusion as to how hard this problem is. Many colleges claim that it is easy, while others complain that it is next to impossible. In the literature, some authors claim or conjecture polynomial time algorithms, while others develop heuristic approaches. The goal of this paper is to resolve the confusion by identifying cases where the problem will be easy and others where it will be hard. We focus on the kinds of cases that schedulers are apt to encounter in practice.