3-approximation Algorithm for the Travelling Repairman Problem with Unit Time-windows Luis Eduardo Urb´ an Rivero 1 , Cynthia A. Rodr´ıguez Villalobos 2 , Rafael L´ opez Bracho 3 , Francisco Javier Zaragoza Mart´ınez 3 1 UAM Azcapotzalco, Posgrado en Optimizaci´ on, Mexico 2 University of Waterloo, Department of Combinatorics and Optimization, Canada 3 UAM Azcapotzalco, Departamento de Sistemas, Mexico lurbanrivero@gmail.com, ca7rodri@uwaterloo.ca, rlb@correo.azc.uam.mx, franz@correo.azc.uam.mx Abstract. The travelling repairman problem (TRP) is a scheduling problem in which a repairman must visit locations to perform some task. Each location has a time window in which the repairman is allowed to arrive. The objective of this problem is to maximize the number of tasks performed. In this paper, we consider a special case in which all the locations are on a straight line, the tasks have no processing time, and all time-windows are of unit length. We present an improvement on the analysis of the 4-approximation algorithm presented by S. L. P´erez P´erez et al. in 2014. Keywords: TRP, approximation algorithm, unit time-windows. 1 Introduction In the travelling repairman problem (TRP), we are given a set of locations the repairman can visit and the time needed to travel between any pair of locations. Each location has a set of tasks to be done by the repairman; each with a fixed processing time and a time-window during which the repairman is allowed to arrive at the location to perform that task. The objective of the problem is to find the route which maximizes the number of tasks completed by the repairman. In 1992, J. Tsitsiklis [5] proved that deciding whether the repairman can complete k tasks within their time-windows is NP-complete, even if the tasks have no processing time. In 2005, R. Bar-Yehuda, et al. [1] considered the special case in which all location are on a line, there are no processing times, and all time-windows are of unit length. They proposed an 8-approximation algorithm with running time 167 Research in Computing Science 107 (2015) pp. 167–173; rec. 2015-08-24; acc. 2015-10-12