A fixed-parameter tractable algorithm for spatio-temporal calendar management

Calendar management tools assist users with coordinating their daily life. Different tasks have to be scheduled according to the user preferences. In many cases, tasks are at different locations and travel times have to be considered. Therefore, these kinds of calendar management problems can be regarded as spatio-temporal optimisation problems and are often variants of traveling salesman problems (TSP) or vehicle routing problems. While standard TSPs require a solution to include all tasks, prize-collecting TSPs are more suited for calendar management problems as they require a solution that optimises the total sum of "prizes" we assigned to tasks at different locations. If we now add time windows that limit when tasks can occur, these prize-collecting TSPs with time windows (TW-TSP) are excellent abstractions of spatio-temporal optimisation problems such as calendar management. Due to the inherent complexity of TW-TSPs, the existing literature considers mainly approximation algorithms or special cases. We present a novel algorithm for TW-TSPs that enables us to find the optimal solution to TW-TSP problems occurring in real-world calendar management applications efficiently. Our algorithm is a fixed-parameter tractable algorithm that depends on the maximal number of tasks that can be revisited from some other task, a parameter which is small in the application scenario we consider.