On a conjecture for the university timetabling problem

The University timetabling is a generalization of the well-known class-teacher timetabling model, where in addition to lectures given by a single teacher to a single class, there are some lectures given by a single teacher to a group of classes simultaneously. One looks for a minimum number of periods in which to complete all lectures without conflicts. The problem is NP-hard in the strong sense even if the number of groups is three, but it is polynomially solvable for two groups. In the latter case, it has been conjectured that the minimum number of periods in which to complete all lectures without conflicts equals ⌈T⌉, where T is the optimal value of an LP-relaxation. The LP-relaxation permits fractions of periods in feasible solutions. We prove this conjecture in this paper.