Online interval scheduling on a single machine with finite lookahead $ Feifeng Zheng a,b , Yongxi Cheng a,n , Ming Liu c , Yinfeng Xu a a School of Management, Xi’an Jiaotong University, Xi’an 710049, China b Glorious Sun School of Business and Management, Donghua University, Shanghai 200051, China c School of Economics & Management, Tongji University, Shanghai 200092, China article info Available online 15 June 2012 Keywords: Online interval scheduling Lookahead Preemption Competitive ratio abstract We study an online weighted interval scheduling problem on a single machine, where all intervals have unit length and the objective is to maximize the total weight of all completed intervals. We investigate how the function of finite lookahead improves the competitivities of deterministic online heuristics, under both preemptive and non-preemptive models. The lookahead model studied in this paper is that an online heuristic is said to have a lookahead ability of LD if at any time point it is able to foresee all the intervals to be released within the next LD units of time. We investigate both competitive online heuristics and lower bounds on the competitive ratio, with lookahead 0 rLD r1 under the preemptive model, and lookahead 0 rLD r2 under the non-preemptive model. A method to transform a preemptive lookahead online algorithm to a non-preemptive online algorithm with enhanced looka- head ability is also given. Computational tests are performed to compare the practical competitivities of the online heuristics with different lookahead abilities. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction In the online single machine interval scheduling problem, there is one machine to schedule a set of weighted intervals with various arrival time, and at any time point at most one interval can be processed on the machine. The goal is to maximize the total weight of all completed intervals. The problem can be viewed as a special job scheduling problem in which each interval is considered to be a job, and each job has, besides its weight, an arrival time and a processing time. All the information of a job becomes known upon its arrival, that is at the release time of the job. If one does not start an interval immediately upon its arrival, or if one aborts an interval before its completion, that interval is lost. The interval scheduling problem arises naturally from various real-life applications, including the assignment of trans- ports to loading/unloading terminals, work planning for person- nel, bandwidth allocation of communication channels, etc. [1]. Refer to [1,2] for recent surveys on offline and online interval scheduling problems and their variants. We use the concept of competitive ratio (see [3]) to measure the performance of an online algorithm A, which is the worst case ratio between the weight obtained by an optimal offline algo- rithm OPT and the weight obtained by A, over all possible input interval sequence I. More specifically, let AðIÞ and I n denote the schedules produced by A and by an optimal offline algorithm OPT, on an input interval sequence I, respectively. Let 9AðIÞ9 and 9I n 9 denote the total weight of all the intervals in AðIÞ and I n , respectively. Then, the competitive ratio of A is defined as c ¼ sup I 9I n 9=9AðIÞ9, where the supremum is taken over all possible input sequence I. If c is finite, then A is said to be competitive, or, to be more specific, c-competitive. For the general online weighted interval scheduling problem, even on a single machine, Woeginger [4] showed that no deter- ministic algorithm has a finite competitive ratio. Later, Canetti and Irani [5] showed that the same also holds for randomized algorithms. On the other hand, randomized competitive algo- rithms do exist for special cases where there is a certain relation between the length of an interval and its weight [6,7]. 1.1. Lookahead and preemption Offline algorithms have all the information of all intervals from the very beginning, while online algorithms know nothing about future intervals. Somewhat in between are the algorithms with certain lookahead ability, which have knowledge of the intervals in the near future. Usually the model with a finite lookahead ability represents a more realistic situation. For example, a doctor who responds to patients’ requests for office visits is unable to know all requests in the future, however, it is possible for him/her Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/caor Computers & Operations Research 0305-0548/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cor.2012.06.003 $ This work was partially supported by the National Natural Science Foundation of China under Grant nos. 71172189, 11101326, 71101106, and 71071123. n Corresponding author. E-mail addresses: zhengff@mail.xjtu.edu.cn (F. Zheng), chengyx@gmail.com, chengyx@mail.xjtu.edu.cn (Y. Cheng), minyivg@gmail.com (M. Liu), yfxu@mail.xjtu.edu.cn (Y. Xu). Computers & Operations Research 40 (2013) 180–191