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.