A convex lasso framework for linear timetabling

Abstract

A new convex optimization framework for approximately solving timetabling problems that can be described as integer linear programs is proposed. The method is based on converting the timetabling problem into a cardinality constrained problem while viewing the operational constraints of the timetable as noisy measurements with an unknown average slack. The problem is then iteratively solved using weighted Lasso with weights that are updated using a simple linear program to satisfy the cardinality constraint while respecting the operational constraints in the least squares sense. Compared with previous convex relaxations for solving timetabling problems, our solution technique will converge to a binary solution without the need for subsequent randomized rounding. Moreover, the number of Lasso iterations required are in the order of the number of events to be scheduled and hence the method can handle very large timetabling problems using efficient Lasso solvers. We provide the assumptions required on the linear timetabling model and establish the associated error bound of the technique upon convergence assuming restricted strong convexity and uniqueness. We also study the effect of the Lasso regularization parameter and the effect of relaxing the objective on the extent of constraint satisfaction through several simulation experiments. Our experiment on one of the benchmark datasets for university examination timetabling improved the best documented result by more than 30% with 99.9% of all constraints satisfied.