A Convex LASSO Framework for Linear Timetabling

Ali Ahmed Al-Matouq

doi:10.1109/access.2020.3019598

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.

Highlights

Timetabling is the assignment of events to a finite number of time-space resources while respecting a set of feasibility and operational constraints and meeting a certain desired objective [1]
A new convex optimization framework for solving automated timetabling problems described as binary linear programs is presented
The method is based on converting the binary linear program into a cardinality constrained problem while relaxing the operational constraints

Summary

INTRODUCTION

Timetabling is the assignment of events to a finite number of time-space resources while respecting a set of feasibility and operational constraints and meeting a certain desired objective [1]. Using exact methods, including cutting plane and/or branch and bound techniques that can be terminated with a certificate proving sub-optimality [14] These techniques can take considerable amount of time and computational resources to solve, especially for timetabling problems that involve very large number of decision variables and constraints [15]. These techniques are usually combined with exact methods like branch and bound and cutting plane approaches and can be used for both linear and nonlinear timetabling problems [27] These techniques can handle large timetabling problems they require randomized rounding techniques after obtaining the solution (similar to SDP relaxations). Our experiments demonstrate that the Lasso regularization parameter need to be sufficiently large to ensure a final binary solution and constraint satisfaction can be significantly improved if a lower bound on the optimal value of the objective is known. In the appendix we present two timetabling examples to demonstrate applicability of the study assumptions to timetabling problems

PROBLEM FORMULATION

REDUCING THE SIZE OF LARGE PROBLEMS

ESTIMATION PERFORMANCE AT CONVERGENCE

SIMULATION STUDY

Findings

CONCLUSION