Online Food Delivery to Minimize Maximum Flow Time

TL;DR

This paper addresses the challenge of minimizing maximum flow time in online food delivery, presenting new algorithms with competitive ratios and exploring the problem's computational hardness.

Contribution

It introduces an $O(1)$-competitive online algorithm for uncapacitated delivery on tree metrics and develops approximation algorithms under speed-augmentation models.

Findings

Hardness of approximation is $oldsymbol{ ilde{oldsymbol{ ext{Omega}}}(n)}$ for offline and online cases.

An $O(1)$-competitive online algorithm exists for uncapacitated delivery on trees.

Approximation algorithms are developed using speed-augmentation with factors related to TSP and CVRP.

Abstract

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.