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Minimizing the Maximum Flow Time in the Online Food Delivery Problem

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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. 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.

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Notes

  1. In FDP, we could define the completion time of a request as the time the vehicle returns to the depot after serving the request. The maximum flow time objective for the two versions differs by an O(1) factor.

  2. Equivalently we could use a metric (V, d) to describe the travel times, but for many of our results it is more convenient to use the graph G with edge lengths.

  3. Notice that the algorithm can easily check if the optimum maximum flow time is 0 or not.

  4. Recall that in the notation \({\textsf{mst}}_e\), we can view requests as vertices by ignoring their arrival times.

  5. By the definition of \(\alpha _{\textsf{TSP}}\) (resp. \(\alpha _{\textsf{CVRP}}\)), we do not know if an \(\alpha _{\textsf{TSP}}\)- (resp. \(\alpha _{\textsf{CVRP}}\)-) approximation for TSP (resp. CVRP) exists or not. So, we need to include an \(\epsilon \) term in the definition of \(\alpha \) in the the second and third cases.

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Acknowledgements

The extended abstract of this paper has been published in Proceedings of the 33rd International Symposium on Algorithms and Computation (ISAAC 2022). KL Luo is supported by the National Natural Science Foundation of China, Grant No. 72071157. YH Zhang is supported by the National Natural Science Foundation of China, Grant No. 62102251.

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Author notes

  1. Xiangyu Guo, Shi Li, Kelin Luo and Yuhao Zhang have contributed equally to this work.

Authors and Affiliations

  1. Department of Computer Science and Engineering, University at Buffalo, 105 White Rd, Buffalo, NY, 14260, USA

    Xiangyu Guo, Shi Li & Kelin Luo

  2. Institute of Computer Science, University of Bonn, Poppelsdorfer Allee 49, 53115, Bonn, NRW, Germany

    Kelin Luo

  3. John Hopcroft Center for Computer Science, Shanghai Jiao Tong University, 800 Dongchuan RD, Shanghai, 200240, SH, China

    Yuhao Zhang

  4. Department of Computer Science and Technology, Nanjing University, Nanjing, China

    Shi Li

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  1. Xiangyu Guo
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Guo, X., Li, S., Luo, K. et al. Minimizing the Maximum Flow Time in the Online Food Delivery Problem. Algorithmica 86, 907–943 (2024). https://doi.org/10.1007/s00453-023-01177-1

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