  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.
  We show that the problem is hard in both offline and online settings: 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. Our main result is an
$O(1)$-competitive online algorithm for the uncapaciated (i.e, $c = \infty$)
food delivery problem on tree metrics. Then we consider the speed-augmentation
model. 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 $\alpha_{TSP}+ \epsilon$ or
$\alpha_{CVRP}+ \epsilon$, depending on whether the problem is uncapacitated.
Here $\alpha_{TSP}$ and $\alpha_{CVRP}$ 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.
