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

A scheduling problem in which jobs have locations and a repairman is supposed to visit customers at their homes is studied, known as the Prize-Collecting Traveling Salesman Problem with time windows (TW-TSP).

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. A geometric interpretation of the problem is used that generalizes the Erdős-Szekeres Theorem. 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.