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**Published in:** LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

We study the unit-demand capacitated vehicle routing problem in the random setting of the Euclidean plane. The objective is to visit n random terminals in a square using a set of tours of minimum total length, such that each tour visits the depot and at most k terminals.
We design an algorithm combining the classical sweep heuristic and the framework for the Euclidean traveling salesman problem due to Arora [J. ACM 1998] and Mitchell [SICOMP 1999]. We show that our algorithm is a polynomial-time approximation of ratio at most 1.55 asymptotically almost surely. This improves on the prior ratio of 1.915 due to Mathieu and Zhou [RSA 2022]. In addition, we conjecture that, for any ε > 0, our algorithm is a (1+ε)-approximation asymptotically almost surely.

Zipei Nie and Hang Zhou. Euclidean Capacitated Vehicle Routing in the Random Setting: A 1.55-Approximation Algorithm. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 91:1-91:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{nie_et_al:LIPIcs.ESA.2024.91, author = {Nie, Zipei and Zhou, Hang}, title = {{Euclidean Capacitated Vehicle Routing in the Random Setting: A 1.55-Approximation Algorithm}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {91:1--91:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.91}, URN = {urn:nbn:de:0030-drops-211627}, doi = {10.4230/LIPIcs.ESA.2024.91}, annote = {Keywords: capacitated vehicle routing, approximation algorithm, combinatorial optimization} }

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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

In the Euclidean k-traveling salesman problem (k-TSP), we are given n points in the d-dimensional Euclidean space, for some fixed constant d ≥ 2, and a positive integer k. The goal is to find a shortest tour visiting at least k points.
We give an approximation scheme for the Euclidean k-TSP in time n⋅2^O(1/ε^{d-1})⋅(log n)^{2d²⋅2^d}. This improves Arora’s approximation scheme of running time n⋅k⋅(log n)^(O(√d/ε))^{d-1}} [J. ACM 1998]. Our algorithm is Gap-ETH tight and can be derandomized by increasing the running time by a factor O(n^d).

Ernest van Wijland and Hang Zhou. Faster Approximation Scheme for Euclidean k-TSP. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 81:1-81:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vanwijland_et_al:LIPIcs.SoCG.2024.81, author = {van Wijland, Ernest and Zhou, Hang}, title = {{Faster Approximation Scheme for Euclidean k-TSP}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {81:1--81:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.81}, URN = {urn:nbn:de:0030-drops-200268}, doi = {10.4230/LIPIcs.SoCG.2024.81}, annote = {Keywords: approximation algorithms, optimization, traveling salesman problem} }

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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)

In the unsplittable capacitated vehicle routing problem (UCVRP) on trees, we are given a rooted tree with edge weights and a subset of vertices of the tree called terminals. Each terminal is associated with a positive demand between 0 and 1. The goal is to find a minimum length collection of tours starting and ending at the root of the tree such that the demand of each terminal is covered by a single tour (i.e., the demand cannot be split), and the total demand of the terminals in each tour does not exceed the capacity of 1.
For the special case when all terminals have equal demands, a long line of research culminated in a quasi-polynomial time approximation scheme [Jayaprakash and Salavatipour, TALG 2023] and a polynomial time approximation scheme [Mathieu and Zhou, TALG 2023].
In this work, we study the general case when the terminals have arbitrary demands. Our main contribution is a polynomial time (1.5+ε)-approximation algorithm for the UCVRP on trees. This is the first improvement upon the 2-approximation algorithm more than 30 years ago. Our approximation ratio is essentially best possible, since it is NP-hard to approximate the UCVRP on trees to better than a 1.5 factor.

Claire Mathieu and Hang Zhou. A Tight (1.5+ε)-Approximation for Unsplittable Capacitated Vehicle Routing on Trees. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 91:1-91:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{mathieu_et_al:LIPIcs.ICALP.2023.91, author = {Mathieu, Claire and Zhou, Hang}, title = {{A Tight (1.5+\epsilon)-Approximation for Unsplittable Capacitated Vehicle Routing on Trees}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {91:1--91:16}, 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.2023.91}, URN = {urn:nbn:de:0030-drops-181430}, doi = {10.4230/LIPIcs.ICALP.2023.91}, annote = {Keywords: approximation algorithms, capacitated vehicle routing, graph algorithms, combinatorial optimization} }

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**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

In the Distance-constrained Vehicle Routing Problem (DVRP), we are given a graph with integer edge weights, a depot, a set of n terminals, and a distance constraint D. The goal is to find a minimum number of tours starting and ending at the depot such that those tours together cover all the terminals and the length of each tour is at most D.
The DVRP on trees is of independent interest, because it is equivalent to the "virtual machine packing" problem on trees studied by Sindelar et al. [SPAA'11]. We design a simple and natural approximation algorithm for the tree DVRP, parameterized by ε > 0. We show that its approximation ratio is α + ε, where α ≈ 1.691, and in addition, that our analysis is essentially tight. The running time is polynomial in n and D. The approximation ratio improves on the ratio of 2 due to Nagarajan and Ravi [Networks'12].
The main novelty of this paper lies in the analysis of the algorithm. It relies on a reduction from the tree DVRP to the bounded space online bin packing problem via a new notion of "reduced length".

Marc Dufay, Claire Mathieu, and Hang Zhou. An Approximation Algorithm for Distance-Constrained Vehicle Routing on Trees. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dufay_et_al:LIPIcs.STACS.2023.27, author = {Dufay, Marc and Mathieu, Claire and Zhou, Hang}, title = {{An Approximation Algorithm for Distance-Constrained Vehicle Routing on Trees}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {27:1--27:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.27}, URN = {urn:nbn:de:0030-drops-176794}, doi = {10.4230/LIPIcs.STACS.2023.27}, annote = {Keywords: vehicle routing, distance constraint, approximation algorithms, trees} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

In the unsplittable capacitated vehicle routing problem, we are given a metric space with a vertex called depot and a set of vertices called terminals. Each terminal is associated with a positive demand between 0 and 1. The goal is to find a minimum length collection of tours starting and ending at the depot such that the demand of each terminal is covered by a single tour (i.e., the demand cannot be split), and the total demand of the terminals in each tour does not exceed the capacity of 1.
Our main result is a polynomial-time (2+ε)-approximation algorithm for this problem in the two-dimensional Euclidean plane, i.e., for the special case where the terminals and the depot are associated with points in the Euclidean plane and their distances are defined accordingly. This improves on recent work by Blauth, Traub, and Vygen [IPCO'21] and Friggstad, Mousavi, Rahgoshay, and Salavatipour [IPCO'22].

Fabrizio Grandoni, Claire Mathieu, and Hang Zhou. Unsplittable Euclidean Capacitated Vehicle Routing: A (2+ε)-Approximation Algorithm. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{grandoni_et_al:LIPIcs.ITCS.2023.63, author = {Grandoni, Fabrizio and Mathieu, Claire and Zhou, Hang}, title = {{Unsplittable Euclidean Capacitated Vehicle Routing: A (2+\epsilon)-Approximation Algorithm}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {63:1--63:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.63}, URN = {urn:nbn:de:0030-drops-175660}, doi = {10.4230/LIPIcs.ITCS.2023.63}, annote = {Keywords: capacitated vehicle routing, approximation algorithms, Euclidean plane} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We give a polynomial time approximation scheme (PTAS) for the unit demand capacitated vehicle routing problem (CVRP) on trees, for the entire range of the tour capacity. The result extends to the splittable CVRP.

Claire Mathieu and Hang Zhou. A PTAS for Capacitated Vehicle Routing on Trees. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 95:1-95:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{mathieu_et_al:LIPIcs.ICALP.2022.95, author = {Mathieu, Claire and Zhou, Hang}, title = {{A PTAS for Capacitated Vehicle Routing on Trees}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {95:1--95:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.95}, URN = {urn:nbn:de:0030-drops-164369}, doi = {10.4230/LIPIcs.ICALP.2022.95}, annote = {Keywords: approximation algorithms, capacitated vehicle routing, graph algorithms, combinatorial optimization} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

We give a probabilistic analysis of the unit-demand Euclidean capacitated vehicle routing problem in the random setting, where the input distribution consists of n unit-demand customers modeled as independent, identically distributed uniform random points in the two-dimensional plane. The objective is to visit every customer using a set of routes of minimum total length, such that each route visits at most k customers, where k is the capacity of a vehicle. All of the following results are in the random setting and hold asymptotically almost surely.
The best known polynomial-time approximation for this problem is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the ITP algorithm is near-optimal when k is either o(√n) or ω(√n), and they asked whether the ITP algorithm was "also effective in the intermediate range". In this work, we show that when k = √n, the ITP algorithm is at best a (1+c₀)-approximation for some positive constant c₀.
On the other hand, the approximation ratio of the ITP algorithm was known to be at most 0.995+α due to Bompadre, Dror, and Orlin, where α is the approximation ratio of an algorithm for the traveling salesman problem. In this work, we improve the upper bound on the approximation ratio of the ITP algorithm to 0.915+α. Our analysis is based on a new lower bound on the optimal cost for the metric capacitated vehicle routing problem, which may be of independent interest.

Claire Mathieu and Hang Zhou. Probabilistic Analysis of Euclidean Capacitated Vehicle Routing. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{mathieu_et_al:LIPIcs.ISAAC.2021.43, author = {Mathieu, Claire and Zhou, Hang}, title = {{Probabilistic Analysis of Euclidean Capacitated Vehicle Routing}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {43:1--43:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.43}, URN = {urn:nbn:de:0030-drops-154769}, doi = {10.4230/LIPIcs.ISAAC.2021.43}, annote = {Keywords: capacitated vehicle routing, iterated tour partitioning, probabilistic analysis, approximation algorithms} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

How efficiently can we find an unknown graph using distance queries between its vertices? We assume that the unknown graph is connected, unweighted, and has bounded degree. The goal is to find every edge in the graph. This problem admits a reconstruction algorithm based on multi-phase Voronoi-cell decomposition and using Õ(n^{3/2}) distance queries [Kannan et al., 2018].
In our work, we analyze a simple reconstruction algorithm. We show that, on random Δ-regular graphs, our algorithm uses Õ(n) distance queries. As by-products, we can reconstruct those graphs using O(log² n) queries to an all-distances oracle or Õ(n) queries to a betweenness oracle, and we bound the metric dimension of those graphs by log² n.
Our reconstruction algorithm has a very simple structure, and is highly parallelizable. On general graphs of bounded degree, our reconstruction algorithm has subquadratic query complexity.

Claire Mathieu and Hang Zhou. A Simple Algorithm for Graph Reconstruction. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 68:1-68:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{mathieu_et_al:LIPIcs.ESA.2021.68, author = {Mathieu, Claire and Zhou, Hang}, title = {{A Simple Algorithm for Graph Reconstruction}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {68:1--68:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.68}, URN = {urn:nbn:de:0030-drops-146496}, doi = {10.4230/LIPIcs.ESA.2021.68}, annote = {Keywords: reconstruction, network topology, random regular graphs, metric dimension} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

In correlation clustering, the input is a graph with edge-weights, where every edge is labelled either + or - according to similarity of its endpoints. The goal is to produce a partition of the vertices that disagrees with the edge labels as little as possible.
In two-edge-connected augmentation, the input is a graph with edge-weights and a subset R of edges of the graph. The goal is to produce a minimum weight subset S of edges of the graph, such that for every edge in R, its endpoints are two-edge-connected in R\cup S.
For planar graphs, we prove that correlation clustering reduces to two-edge-connected augmentation, and that both problems have a polynomial-time approximation scheme.

Philip N. Klein, Claire Mathieu, and Hang Zhou. Correlation Clustering and Two-edge-connected Augmentation for Planar Graphs. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 554-567, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{klein_et_al:LIPIcs.STACS.2015.554, author = {Klein, Philip N. and Mathieu, Claire and Zhou, Hang}, title = {{Correlation Clustering and Two-edge-connected Augmentation for Planar Graphs}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {554--567}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.554}, URN = {urn:nbn:de:0030-drops-49411}, doi = {10.4230/LIPIcs.STACS.2015.554}, annote = {Keywords: correlation clustering, two-edge-connected augmentation, polynomial-time approximation scheme, planar graphs} }

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