Learning-augmented Rent-or-Buy with a Sample

In this paper, we study the rent-or-buy problem (also called the Bahncard problem) in the learning-augmented setting. In this problem, a traveler must complete a sequence of trips that are revealed online over time, each of which has an associated cost with it. The traveler has the option of buying a discount card at a fixed cost that gives a discount on trip costs for a fixed time after buying the card. The goal is to minimize the overall cost of all the trips, including the money spent on buying discount cards. For this problem, it is well-known that the best deterministic algorithm has a competitive ratio of 2. In this paper, we ask whether we can do better if the traveler has a sample of trips available offline, e.g., obtained from an ML model based on historical data. We show that even a sparse sample of the input can significantly improve the competitive ratio of the algorithm from 2 to 3/2, and further to close to 1 under some additional conditions. We also verify our theoretical bounds via numerical simulations, which reveal that our proposed algorithm obtains nearly optimal solutions for a variety of natural input classes.