Minimizing the Maximum Flow Time in the Online Food Delivery Problem

Authors Xiangyu Guo, Kelin Luo , Shi Li, Yuhao Zhang

Thumbnail PDF


  • Filesize: 0.88 MB
  • 18 pages

Document Identifiers

Author Details

Xiangyu Guo
  • University at Buffalo, NY, USA
Kelin Luo
  • Institute of Computer Science, Universität Bonn, Germany
Shi Li
  • University at Buffalo, NY, USA
Yuhao Zhang
  • Shanghai Jiao Tong University, China

Cite AsGet BibTex

Xiangyu Guo, Kelin Luo, Shi Li, and Yuhao Zhang. Minimizing the Maximum Flow Time in the Online Food Delivery Problem. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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 even when k = 1 and c = ∞: There is a hardness of approximation of Ω(n) for the offline problem, and a lower bound of Ω(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 uncapacitated (i.e, c = ∞) food delivery problem on tree metrics; we also have negative result showing that the condition c = ∞ is needed. Then we explore the speed-augmentation model where our online algorithm is allowed to use vehicles with faster speed. We show that a moderate speeding factor leads to a constant competitive ratio, and we prove a tight trade-off between the speeding factor and the competitive ratio.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Online algorithm
  • Capacitated Vehicle Routing
  • Flow Time Optimization


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Kemal Altinkemer and Bezalel Gavish. Technical note - heuristics for delivery problems with constant error guarantees. Transportation Science, 24:294-297, 1990. Google Scholar
  2. Norbert Ascheuer, Sven O Krumke, and Jörg Rambau. Online dial-a-ride problems: Minimizing the completion time. In Annual Symposium on Theoretical Aspects of Computer Science, pages 639-650. Springer, 2000. Google Scholar
  3. Nikhil Bansal and Ho-Leung Chan. Weighted flow time does not admit o(1)-competitive algorithms. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'09, pages 1238-1244, USA, 2009. Society for Industrial and Applied Mathematics. Google Scholar
  4. Nikhil Bansal, Ho-Leung Chan, Rohit Khandekar, Kirk Pruhs, Baruch Schieber, and Cliff Stein. Non-preemptive min-sum scheduling with resource augmentation. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07), pages 614-624. IEEE, 2007. Google Scholar
  5. Nikhil Bansal, Moses Charikar, Sanjeev Khanna, and Joseph Naor. Approximating the average response time in broadcast scheduling. In SODA, volume 5, pages 215-221. Citeseer, 2005. Google Scholar
  6. Nikhil Bansal and Bouke Cloostermans. Minimizing maximum flow-time on related machines. Theory of Computing, 12(1):1-14, 2016. Google Scholar
  7. Yair Bartal and Shan Muthukrishnan. Minimizing maximum response time in scheduling broadcasts. In Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms, pages 558-559, 2000. Google Scholar
  8. Jatin Batra, Naveen Garg, and Amit Kumar. Constant factor approximation algorithm for weighted flow time on a single machine in pseudo-polynomial time. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 778-789. IEEE Computer Society, 2018. Google Scholar
  9. Marcin Bienkowski, Artur Kraska, and Hsiang-Hsuan Liu. Traveling repairperson, unrelated machines, and other stories about average completion times. In International Colloquium on Automata, Languages and Programming (ICALP), 2021. Google Scholar
  10. Jannis Blauth, Vera Traub, and Jens Vygen. Improving the approximation ratio for capacitated vehicle routing. In International Conference on Integer Programming and Combinatorial Optimization, pages 1-14. Springer, 2021. Google Scholar
  11. Jessica Chang, Thomas Erlebach, Renars Gailis, and Samir Khuller. Broadcast scheduling: algorithms and complexity. ACM Transactions on Algorithms (TALG), 7(4):1-14, 2011. Google Scholar
  12. Moses Charikar and Balaji Raghavachari. The finite capacity dial-a-ride problem. In Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No. 98CB36280), pages 458-467. IEEE, 1998. Google Scholar
  13. Chandra Chekuri, Sungjin Im, and Benjamin Moseley. Minimizing maximum response time and delay factor in broadcast scheduling. In European Symposium on Algorithms, pages 444-455. Springer, 2009. Google Scholar
  14. Anamitra Roy Choudhury, Syamantak Das, Naveen Garg, and Amit Kumar. Rejecting jobs to minimize load and maximum flow-time. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1114-1133. SIAM, 2014. Google Scholar
  15. Vincent Cohen-Addad, Arnold Filtser, Philip N Klein, and Hung Le. On light spanners, low-treewidth embeddings and efficient traversing in minor-free graphs. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 589-600. IEEE, 2020. Google Scholar
  16. Aparna Das and Claire Mathieu. A quasi-polynomial time approximation scheme for euclidean capacitated vehicle routing. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 390-403. SIAM, 2010. Google Scholar
  17. Leah Epstein and Rob van Stee. Optimal on-line flow time with resource augmentation. Discrete applied mathematics, 154(4):611-621, 2006. Google Scholar
  18. Uriel Feige, Janardhan Kulkarni, and Shi Li. A polynomial time constant approximation for minimizing total weighted flow-time. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1585-1595. SIAM, 2019. Google Scholar
  19. Esteban Feuerstein and Leen Stougie. On-line single-server dial-a-ride problems. Theoretical Computer Science, 268(1):91-105, 2001. Google Scholar
  20. Inge Li Gørtz. Hardness of preemptive finite capacity dial-a-ride. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 200-211. Springer, 2006. Google Scholar
  21. Inge Li Gørtz, Viswanath Nagarajan, and R Ravi. Minimum makespan multi-vehicle dial-a-ride. In European Symposium on Algorithms, pages 540-552. Springer, 2009. Google Scholar
  22. Xiangyu Guo, Kelin Luo, Zhihao Gavin Tang, and Yuhao Zhang. The online food delivery problem on stars. Theoretical Computer Science, 2022. URL:
  23. Anupam Gupta, MohammadTaghi Hajiaghayi, Viswanath Nagarajan, and Ramamoorthi Ravi. Dial a ride from k-forest. ACM Transactions on Algorithms (TALG), 6(2):1-21, 2010. Google Scholar
  24. Mordecai Haimovich and Alexander HG Rinnooy Kan. Bounds and heuristics for capacitated routing problems. Mathematics of operations Research, 10(4):527-542, 1985. Google Scholar
  25. Dawsen Hwang and Patrick Jaillet. Online scheduling with multi-state machines. Networks, 71(3):209-251, 2018. Google Scholar
  26. Sungjin Im, Shi Li, Benjamin Moseley, and Eric Torng. A dynamic programming framework for non-preemptive scheduling problems on multiple machines. In Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms, pages 1070-1086. SIAM, 2014. Google Scholar
  27. Aditya Jayaprakash and Mohammad R Salavatipour. Approximation schemes for capacitated vehicle routing on graphs of bounded treewidth, bounded doubling, or highway dimension. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 877-893. SIAM, 2022. Google Scholar
  28. Bala Kalyanasundaram and Kirk Pruhs. Speed is as powerful as clairvoyance. Journal of the ACM (JACM), 47(4):617-643, 2000. Google Scholar
  29. Bala Kalyanasundaram, Kirk Pruhs, and Mahe Velauthapillai. Scheduling broadcasts in wireless networks. In European Symposium on Algorithms, pages 290-301. Springer, 2000. Google Scholar
  30. Anna R Karlin, Nathan Klein, and Shayan Oveis Gharan. A (slightly) improved approximation algorithm for metric tsp. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 32-45, 2021. Google Scholar
  31. Hans Kellerer, Thomas Tautenhahn, and Gerhard J. Woeginger. Approximability and nonapproximability results for minimizing total flow time on a single machine. In Gary L. Miller, editor, Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pages 418-426. ACM, 1996. URL:
  32. Michael Khachay and Roman Dubinin. Ptas for the euclidean capacitated vehicle routing problem in r^d. In International Conference on Discrete Optimization and Operations Research, pages 193-205. Springer, 2016. Google Scholar
  33. Sven O Krumke, Willem E de Paepe, Diana Poensgen, Maarten Lipmann, Alberto Marchetti-Spaccamela, and Leen Stougie. On minimizing the maximum flow time in the online dial-a-ride problem. In International Workshop on Approximation and Online Algorithms, pages 258-269. Springer, 2005. Google Scholar
  34. Sven O Krumke, Willem E De Paepe, Diana Poensgen, and Leen Stougie. News from the online traveling repairman. Theoretical Computer Science, 295(1-3):279-294, 2003. Google Scholar
  35. Sven Oliver Krumke, Luigi Laura, Maarten Lipmann, Alberto Marchetti-Spaccamela, Willem de Paepe, Diana Poensgen, and Leen Stougie. Non-abusiveness helps: An o(1)-competitive algorithm for minimizing the maximum flow time in the online traveling salesman problem. In APPROX, pages 200-214. Springer, 2002. Google Scholar
  36. Martine Labbé, Gilbert Laporte, and Hélene Mercure. Capacitated vehicle routing on trees. Operations Research, 39(4):616-622, 1991. Google Scholar
  37. Giorgio Lucarelli, Benjamin Moseley, Nguyen Kim Thang, Abhinav Srivastav, and Denis Trystram. Online non-preemptive scheduling on unrelated machines with rejections. In Proceedings of the 30th on Symposium on Parallelism in Algorithms and Architectures, pages 291-300, 2018. Google Scholar
  38. Giorgio Lucarelli, Benjamin Moseley, Nguyen Kim Thang, Abhinav Srivastav, and Denis Trystram. Online non-preemptive scheduling to minimize weighted flow-time on unrelated machines. arXiv preprint, 2018. URL:
  39. Giorgio Lucarelli, Nguyen Kim Thang, Abhinav Srivastav, and Denis Trystram. Online non-preemptive scheduling in a resource augmentation model based on duality. In European Symposium on Algorithms (ESA 2016), volume 57(63), pages 1-17, 2016. Google Scholar
  40. Monaldo Mastrolilli. Scheduling to minimize max flow time: Off-line and on-line algorithms. International Journal of Foundations of Computer Science, 15(02):385-401, 2004. Google Scholar
  41. Claire Mathieu and Hang Zhou. A ptas for capacitated vehicle routing on trees. arXiv preprint, 2021. URL:
  42. Cynthia A Phillips, Cliff Stein, Eric Torng, and Joel Wein. Optimal time-critical scheduling via resource augmentation. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pages 140-149, 1997. Google Scholar
  43. Lars Rohwedder and Andreas Wiese. A (2+ε)-approximation algorithm for preemptive weighted flow time on a single machine. arXiv preprint, 2020. URL:
  44. Yuanxiao Wu and Xiwen Lu. Capacitated vehicle routing problem on line with unsplittable demands. Journal of Combinatorial Optimization, pages 1-11, 2020. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail