Minimizing the Maximum Flow Time in the Online Food Delivery Problem

We study a common delivery problem encountered in nowadays online food-ordering platforms: Customers order dishes online, and the restaurant delivers the food after receiving the order. Specifically, we study a problem where k vehicles of capacity c are serving a set of requests ordering food from one restaurant. After a request arrives, it can be served by a vehicle moving from the restaurant to its delivery location. We are interested in serving all requests while minimizing the maximum flow-time, i.e., the maximum time length a customer waits to receive his/her food after submitting the order. The problem also has a close connection with the broadcast scheduling problem with maximum flow time objective. We show that the problem is hard in both offline and online settings even when \(k = 1\) and \(c = \infty \): There is a hardness of approximation of \(\Omega (n)\) for the offline problem, and a lower bound of \(\Omega (n)\) on the competitive ratio of any online algorithm, where n is number of points in the metric. We circumvent the strong negative results in two directions. Our main result is an O(1)-competitive online algorithm for the uncapaciated (i.e, \(c = \infty \)) food delivery problem on tree metrics; we also have a negative result showing that the condition \(c = \infty \) is needed. Then we consider the speed-augmentation model, in which our online algorithm is allowed to use \(\alpha \)-speed vehicles, where \(\alpha \ge 1\) is called the speeding factor. We develop an exponential time \((1+\epsilon )\)-speeding \(O(1/\epsilon )\)-competitive algorithm for any \(\epsilon > 0\). A polynomial time algorithm can be obtained with a speeding factor of \(\alpha _{\textsf{TSP}}+ \epsilon \) or \(\alpha _{\textsf{CVRP}}+ \epsilon \), depending on whether the problem is uncapacitated. Here \(\alpha _{\textsf{TSP}}\) and \(\alpha _{\textsf{CVRP}}\) are the best approximation factors for the traveling salesman (TSP) and capacitated vehicle routing (CVRP) problems respectively. We complement the results with some negative ones.