Optimal University Course Timetables and the Partial Transversal Polytope

Abstract

University course timetabling is the conflict-free assignment of courses to weekly time slots and rooms subject to various hard and soft constraints. One goal is to meet as closely as possible professors' preferences. Building on an intuitive integer program (IP), we develop an exact decomposition approach which schedules courses first, and matches courses/times to rooms in a second stage. The subset of constraints which ensures a feasible room assignment defines the well-known partial transversal polytope. We describe it as a polymatroid, and thereby obtain a complete characterization of its facets. This enables us to add only strong valid inequalities to the first stage IP. In fact, for all practical purposes the number of facets is small. We present encouraging computational results on real-world and simulated timetabling data. The sizes of our optimally solvable instances (respecting all hard constraints) are the largest reported in the literature by far.

Curriculum based course timetabling: new solutions to Udine benchmark instances

“…The resulting integer program obviously has an exponential number of constraints (3), and we give details in Lach and Lübbecke (2008) on how to cope with this (and why in practice there are not too many of them).…”

“…Our approach was originally meant to solve instances from Berlin's technical university where all constraints are considered hard, see Lach and Lübbecke (2008). However, feedback on that approach motivated us to evaluate its suitability for incorporating soft constraints, or to obtain lower bounds.…”

“…In a companion paper (Lach and Lübbecke 2008) we developed the theoretical background for our decomposition which considered hard constraints only. Here, we report on how to make it useful in practice, in particular, we state how to incorporate a variety of soft constraints.…”

A Two-Stage Decomposition of High School Timetabling applied to cases in Denmark

“…Therefore this paper shows a different type of decomposition, a Two-Stage Decomposition (TSD). Such an approach was first used for timetabling applications in Lach and Lübbecke (2008) and Lach and Lübbecke (2012) with great success for the curriculum-based university course timetabling problem. The goal of this paper will be to modify the aforementioned approach to be applicable for the high school timetabling problem -giving special attention to the high school system in Denmark.…”

“…Therefore the overall benefits of the TSD out-weight the downsides, and computational results will show that it is indeed way more effective than solving the original IP. The contributions of this paper are the following: 1) It is shown that the approach from Lach and Lübbecke (2008) can also be applied to a high school timetabling problem originating from a practical setting, and by extensive computational results it is argued that the TSD is more effective than solving the original IP. Notice that a similar composition is briefly mentioned in Sørensen and Stidsen (2013) for the same high school timetabling problem, but this paper enhances the approach such that the theoretical maximum gap from optimality is narrowed.…”