Invited Review Fixed interval scheduling: Models, applications, computational complexity and algorithms Mikhail Y. Kovalyov a, * , C.T. Ng b , T.C. Edwin Cheng b a Faculty of Economics, Belarus State University, and United Institute of Informatics Problems, National Academy of Sciences of Belarus, Skorini 4, 220050 Minsk, Belarus b Department of Logistics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Received 5 April 2005; accepted 24 January 2006 Available online 7 July 2006 Abstract The defining characteristic of fixed interval scheduling problems is that each job has a finite number of fixed processing intervals. A job can be processed only in one of its intervals on one of the available machines, or is not processed at all. A decision has to be made about a subset of the jobs to be processed and their assignment to the processing intervals such that the intervals on the same machine do not intersect. These problems arise naturally in different real-life operations plan- ning situations, including the assignment of transports to loading/unloading terminals, work planning for personnel, com- puter wiring, bandwidth allocation of communication channels, printed circuit board manufacturing, gene identification and examining computer memory structures. We present a general formulation of the interval scheduling problem, show its relations to cognate problems in graph theory, and survey existing models, results on computational complexity and solution algorithms. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Interval scheduling; Interval graph; k-coloring; k-track assignment; Discrete starting times; Maximum weight clique; Maxi- mum weight independent set 1. General formulation of the fixed interval scheduling problem Various formulations and special cases of the problem to be addressed in this paper have been considered in the literature under different labels, including ‘‘(fixed) interval scheduling’’, ‘‘interval selection’’, ‘‘scheduling with discrete starting times’’, ‘‘fixed job scheduling’’, ‘‘channel assignment (reservation)’’, ‘‘bandwidth allocation’’, ‘‘k-track assignment’’, ‘‘k-coloring of intervals’’, ‘‘finding K- independent sets on interval graphs’’, ‘‘on-line interval scheduling’’, ‘‘seat reservation’’ and ‘‘maxi- mizing the number of on-time jobs’’, among others. We use the term ‘‘fixed interval scheduling’’ because it appears to be the most popular in the literature and adequately reflects the nature of the underlying problems. The results for several variants of the 0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.01.049 * Corresponding author. Tel.: +375 17 2842125; fax: +375 17 2318403. E-mail addresses: koval@newman.bas-net.by (M.Y. Kova- lyov), lgtctng@polyu.edu.hk (C.T. Ng), lgtcheng@polyu.edu.hk (T.C. Edwin Cheng). European Journal of Operational Research 178 (2007) 331–342 www.elsevier.com/locate/ejor