On approximating a geometric prize-collecting traveling salesman problem with time windows

We study a scheduling problem in which jobs have locations. For example, consider a repairman that is supposed to visit customers at their homes. Each customer is given a time window during which the repairman is allowed to arrive. The goal is to find a schedule that visits as many homes as possible. We refer to this problem as the prize-collecting traveling salesman problem with time windows (TW-TSP). We consider two versions of TW-TSP. In the first version, jobs are located on a line, have release times and deadlines but no processing times. We present a geometric interpretation of TW-TSP on a line that generalizes the longest monotone subsequence problem. We present an O(logn) approximation algorithm for this case, where n denotes the number of jobs. This algorithm can be extended to deal with non-unit job profits. The second version deals with a general case of asymmetric distances between locations. We define a density parameter that, loosely speaking, bounds the number of zig-zags between locations within a time window. We present a dynamic programming algorithm that finds a tour that visits at least OPT/density locations during their time windows. This algorithm can be extended to deal with non-unit job profits and processing times.  2003 Elsevier Inc. All rights reserved. ✩ An extended abstract of this paper appeared in 11th Annual European Symposium on Algorithms, in: Lecture Notes in Comput. Sci., vol. 2832, Springer-Verlag, 2003, pp. 55–66. * Corresponding author. E-mail addresses: [email protected] (R. Bar-Yehuda), [email protected] (G. Even), [email protected] (S. Shahar). 0196-6774/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgor.2003.11.002 R. Bar-Yehuda et al. / Journal of Algorithms 55 (2005) 76–92 77