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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Motivated by an application from geodesy, we study the connected k-center problem and the connected k-diameter problem. These problems arise from the classical k-center and k-diameter problems by adding a side constraint. For the side constraint, we are given an undirected connectivity graph G on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in G. Usually in clustering problems one assumes that the clusters are pairwise disjoint. We study this case but additionally also the case that clusters are allowed to be non-disjoint. This can help to satisfy the connectivity constraints.
Our main result is an O(1)-approximation algorithm for the disjoint connected k-center and k-diameter problem for Euclidean spaces of low dimension (constant d) and for metrics with constant doubling dimension. For general metrics, we get an O(log²k)-approximation. Our algorithms work by computing a non-disjoint connected clustering first and transforming it into a disjoint connected clustering.
We complement these upper bounds by several upper and lower bounds for variations and special cases of the model.

Lukas Drexler, Jan Eube, Kelin Luo, Heiko Röglin, Melanie Schmidt, and Julian Wargalla. Connected k-Center and k-Diameter Clustering. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 50:1-50:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{drexler_et_al:LIPIcs.ICALP.2023.50, author = {Drexler, Lukas and Eube, Jan and Luo, Kelin and R\"{o}glin, Heiko and Schmidt, Melanie and Wargalla, Julian}, title = {{Connected k-Center and k-Diameter Clustering}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {50:1--50:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.50}, URN = {urn:nbn:de:0030-drops-181024}, doi = {10.4230/LIPIcs.ICALP.2023.50}, annote = {Keywords: Approximation algorithms, Clustering, Connectivity constraints} }

Document

**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 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.

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)

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@InProceedings{guo_et_al:LIPIcs.ISAAC.2022.33, author = {Guo, Xiangyu and Luo, Kelin and Li, Shi and Zhang, Yuhao}, title = {{Minimizing the Maximum Flow Time in the Online Food Delivery Problem}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {33:1--33:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.33}, URN = {urn:nbn:de:0030-drops-173181}, doi = {10.4230/LIPIcs.ISAAC.2022.33}, annote = {Keywords: Online algorithm, Capacitated Vehicle Routing, Flow Time Optimization} }

Document

**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

For the problem of delivering a package from a source node to a destination node in a graph using a set of drones, we study the setting where the movements of each drone are restricted to a certain subgraph of the given graph. We consider the objectives of minimizing the delivery time (problem DDT) and of minimizing the total energy consumption (problem DDC). For general graphs, we show a strong inapproximability result and a matching approximation algorithm for DDT as well as NP-hardness and a 2-approximation algorithm for DDC. For the special case of a path, we show that DDT is NP-hard if the drones have different speeds. For trees, we give optimal algorithms under the assumption that all drones have the same speed or the same energy consumption rate. The results for trees extend to arbitrary graphs if the subgraph of each drone is isometric.

Thomas Erlebach, Kelin Luo, and Frits C.R. Spieksma. Package Delivery Using Drones with Restricted Movement Areas. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 49:1-49:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{erlebach_et_al:LIPIcs.ISAAC.2022.49, author = {Erlebach, Thomas and Luo, Kelin and Spieksma, Frits C.R.}, title = {{Package Delivery Using Drones with Restricted Movement Areas}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {49:1--49:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.49}, URN = {urn:nbn:de:0030-drops-173343}, doi = {10.4230/LIPIcs.ISAAC.2022.49}, annote = {Keywords: Mobile agents, approximation algorithm, inapproximability} }

Document

Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

A temporal graph is a graph whose edge set can change over time. We only require that the edge set in each time step forms a connected graph. The temporal exploration problem asks for a temporal walk that starts at a given vertex, moves over at most one edge in each time step, visits all vertices, and reaches the last unvisited vertex as early as possible. We show in this paper that every temporal graph with n vertices can be explored in O(n^{1.75}) time steps provided that either the degree of the graph is bounded in each step or the temporal walk is allowed to make two moves per step. This result is interesting because it breaks the lower bound of Omega(n^2) steps that holds for the worst-case exploration time if only one move per time step is allowed and the graph in each step can have arbitrary degree. We complement this main result by a logarithmic inapproximability result and a proof that for sparse temporal graphs (i.e., temporal graphs with O(n) edges in the underlying graph) making O(1) moves per time step can improve the worst-case exploration time at most by a constant factor.

Thomas Erlebach, Frank Kammer, Kelin Luo, Andrej Sajenko, and Jakob T. Spooner. Two Moves per Time Step Make a Difference. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 141:1-141:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{erlebach_et_al:LIPIcs.ICALP.2019.141, author = {Erlebach, Thomas and Kammer, Frank and Luo, Kelin and Sajenko, Andrej and Spooner, Jakob T.}, title = {{Two Moves per Time Step Make a Difference}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {141:1--141:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.141}, URN = {urn:nbn:de:0030-drops-107176}, doi = {10.4230/LIPIcs.ICALP.2019.141}, annote = {Keywords: Temporal Graph Exploration, Algorithmic Graph Theory, NP-Complete Problem} }

Document

**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

We study an on-line scheduling problem that is motivated by applications such as car-sharing for trips between an airport and a group of hotels. Users submit ride requests, and the scheduler aims to accept requests of maximum total profit using k servers (cars). Each ride request specifies the pick-up time, the pick-up location, and the drop-off location, where one of the two locations must be the airport. A request must be submitted a fixed amount of time before the pick-up time. The scheduler has to decide whether or not to accept a request immediately at the time when the request is submitted (booking time). In the unit travel time variant, the travel time between the airport and any hotel is a fixed value t. We give a 2-competitive algorithm for the case in which the booking interval (pick-up time minus booking time) is at least t and the number of servers is even. In the arbitrary travel time variant, the travel time between the airport and a hotel may have arbitrary length between t and L t for some L >= 1. We give an algorithm with competitive ratio O(log L) if the number of servers is at least ceil[log L]. For both variants, we prove matching lower bounds on the competitive ratio of any deterministic on-line algorithm.

Kelin Luo, Thomas Erlebach, and Yinfeng Xu. Car-Sharing on a Star Network: On-Line Scheduling with k Servers. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 51:1-51:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{luo_et_al:LIPIcs.STACS.2019.51, author = {Luo, Kelin and Erlebach, Thomas and Xu, Yinfeng}, title = {{Car-Sharing on a Star Network: On-Line Scheduling with k Servers}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {51:1--51:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.51}, URN = {urn:nbn:de:0030-drops-102907}, doi = {10.4230/LIPIcs.STACS.2019.51}, annote = {Keywords: Car-Sharing System, On-Line Scheduling, Competitive Analysis, Star Network} }

Document

**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We study an on-line scheduling problem that is motivated by applications such as car-sharing, in which users submit ride requests, and the scheduler aims to accept requests of maximum total profit using k servers (cars). Each ride request specifies the pick-up time and the pick-up location (among two locations, with the other location being the destination). The scheduler has to decide whether or not to accept a request immediately at the time when the request is submitted (booking time). We consider two variants of the problem with respect to constraints on the booking time: In the fixed booking time variant, a request must be submitted a fixed amount of time before the pick-up time. In the variable booking time variant, a request can be submitted at any time during a certain time interval (called the booking horizon) that precedes the pick-up time. We present lower bounds on the competitive ratio for both variants and propose a balanced greedy algorithm (BGA) that achieves the best possible competitive ratio. We prove that, for the fixed booking time variant, BGA is 1.5-competitive if k=3i ( i in N) and the fixed booking length is not less than the travel time between the two locations; for the variable booking time variant, BGA is 1.5-competitive if k=3i ( i in N) and the length of the booking horizon is less than the travel time between the two locations, and BGA is 5/3-competitive if k=5i ( i in N) and the length of the booking horizon is not less than the travel time between the two locations.

Kelin Luo, Thomas Erlebach, and Yinfeng Xu. Online Scheduling of Car-Sharing Requests Between Two Locations with Many Cars and Flexible Advance Bookings. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 64:1-64:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{luo_et_al:LIPIcs.ISAAC.2018.64, author = {Luo, Kelin and Erlebach, Thomas and Xu, Yinfeng}, title = {{Online Scheduling of Car-Sharing Requests Between Two Locations with Many Cars and Flexible Advance Bookings}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {64:1--64:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.64}, URN = {urn:nbn:de:0030-drops-100122}, doi = {10.4230/LIPIcs.ISAAC.2018.64}, annote = {Keywords: Car-sharing system, Competitive analysis, On-line scheduling} }

Document

**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

In this paper, we consider an on-line scheduling problem that is motivated by applications such as car sharing, in which users submit ride requests, and the scheduler aims to accept requests of maximum total profit using two servers (cars). Each ride request specifies the pick-up time and the pick-up location (among two locations, with the other location being the destination). The length of the time interval between the submission of a request (booking time) and the pick-up time is fixed. The scheduler has to decide whether or not to accept a request immediately at the time when the request is submitted. We present lower bounds on the competitive ratio for this problem and propose a smart greedy algorithm that achieves the best possible competitive ratio.

Kelin Luo, Thomas Erlebach, and Yinfeng Xu. Car-Sharing between Two Locations: Online Scheduling with Two Servers. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 50:1-50:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{luo_et_al:LIPIcs.MFCS.2018.50, author = {Luo, Kelin and Erlebach, Thomas and Xu, Yinfeng}, title = {{Car-Sharing between Two Locations: Online Scheduling with Two Servers}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {50:1--50:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.50}, URN = {urn:nbn:de:0030-drops-96325}, doi = {10.4230/LIPIcs.MFCS.2018.50}, annote = {Keywords: Car-sharing system, Competitive analysis, On-line scheduling} }

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