On Approximating a Geometric Prize-Collecting Traveling Salesman Problem with Time Windows: Extended Abstract

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(log n) 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. An extended abstract of this paper appeared in 11th Annual European Symposium on Algorithms, Lecture Notes in Computer Science Volume 2832, pp. 55 66, Springer-Verlag, Sept. 2003. Computer Science Dept., Technion, Haifa 32000, Israel. E-mail:[email protected]. Dept. of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel. E-mail:[email protected]. Dept. of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel. E-mail:[email protected].