Metric $k$-center

Mon, 07 Jul 2025 00:00:00 +0000

General $k$-center problem statement: Let $(X, d)$ be a metric space where $X$ is a set and $d$ is a metric. A set $V \subseteq X$ is provided together with a parameter $k$. The goal is to find a subset $C \subseteq V$ with $|C| = k$ such that the maximum distance of a point in $V$ to the closest point in $C$ is minimized. The problem can be formally defined as follows:

Input: a set $V \subseteq X$, and a parameter $k$.
Output: a set $C \subseteq V$ of $k$ points.
Goal: Minimize the cost $r^C(V) = \max_{v \in V} d(v, C)$

The k-Center Clustering problem can also be defined on a complete undirected graph $G = (V, E)$ as follows:

The $k$-Center Clustering problem: Given a complete undirected graph $G = (V, E)$ with distances $d(v_i, v_j) \in \mathbb{N}$ satisfying the triangle inequality, find a subset $C \subseteq V$ with $|C| = k$ while minimizing:

$$ \max_{v \in V} \min_{c \in C} d(v, c) $$

Mathematics - Combinatorics

Thu, 27 Jun 2024 23:14:15 +0800

Related fields:

Combinatorial optimization
Coding theory
Discrete and computational geometry
Combinatorics and dynamical systems
Combinatorics and physics

List of Selected Papers on Algorithms for Large-Scale Graph Processing.

Sat, 19 Aug 2023 00:00:00 +0000

1/ [ISAAC'11] Goodrich, M. T., Sitchinava, N., & Zhang, Q. (2011, December). Sorting, searching, and simulation in the mapreduce framework. In International Symposium on Algorithms and Computation (pp. 374-383). Springer, Berlin, Heidelberg.

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@inproceedings{goodrich2011sorting,
 title={Sorting, searching, and simulation in the mapreduce framework},
 author={Goodrich, Michael T and Sitchinava, Nodari and Zhang, Qin},
 booktitle={International Symposium on Algorithms and Computation},
 pages={374--383},
 year={2011},
 organization={Springer}
}

2/ [STOC'14] Andoni, A., Nikolov, A., Onak, K., & Yaroslavtsev, G. (2014, May). Parallel algorithms for geometric graph problems. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing (pp. 574-583).

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@inproceedings{andoni2014parallel,
 title={Parallel algorithms for geometric graph problems},
 author={Andoni, Alexandr and Nikolov, Aleksandar and Onak, Krzysztof and Yaroslavtsev, Grigory},
 booktitle={Proceedings of the forty-sixth annual ACM symposium on Theory of computing},
 pages={574--583},
 year={2014}
}

3/ [STOC'17] Im, S., Moseley, B., & Sun, X. (2017, June). Efficient massively parallel methods for dynamic programming. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (pp. 798-811).

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@inproceedings{im2017efficient,
 title={Efficient massively parallel methods for dynamic programming},
 author={Im, Sungjin and Moseley, Benjamin and Sun, Xiaorui},
 booktitle={Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing},
 pages={798--811},
 year={2017}
}

4/ [FOCS'18] Andoni, A., Song, Z., Stein, C., Wang, Z., & Zhong, P. (2018, October). Parallel graph connectivity in log diameter rounds. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (pp. 674-685). IEEE.

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@inproceedings{andoni2018parallel,
 title={Parallel graph connectivity in log diameter rounds},
 author={Andoni, Alexandr and Song, Zhao and Stein, Clifford and Wang, Zhengyu and Zhong, Peilin},
 booktitle={2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)},
 pages={674--685},
 year={2018},
 organization={IEEE}
}

5/ [SOSA'19] Liu, P., & Vondrák, J. (2018). Submodular optimization in the mapreduce model. arXiv preprint arXiv:1810.01489.

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@article{liu2018submodular,
 title={Submodular optimization in the mapreduce model},
 author={Liu, Paul and Vondr{\'a}k, Jan},
 journal={arXiv preprint arXiv:1810.01489},
 year={2018}
}

6/ [PODC'19] Behnezhad, S., Brandt, S., Derakhshan, M., Fischer, M., Hajiaghayi, M., Karp, R. M., & Uitto, J. (2019, July). Massively parallel computation of matching and MIS in sparse graphs. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing (pp. 481-490).

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@inproceedings{behnezhad2019massively,
 title={Massively parallel computation of matching and MIS in sparse graphs},
 author={Behnezhad, Soheil and Brandt, Sebastian and Derakhshan, Mahsa and Fischer, Manuela and Hajiaghayi, MohammadTaghi and Karp, Richard M and Uitto, Jara},
 booktitle={Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing},
 pages={481--490},
 year={2019}
}

Brandt, S., Fischer, M., & Uitto, J. (2018). Matching and MIS for uniformly sparse graphs in the low-memory MPC model. arXiv preprint arXiv:1807.05374.

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@article{brandt2018matching,
 title={Matching and MIS for uniformly sparse graphs in the low-memory MPC model},
 author={Brandt, Sebastian and Fischer, Manuela and Uitto, Jara},
 journal={arXiv preprint arXiv:1807.05374},
 year={2018}
}

Behnezhad, S., Derakhshan, M., Hajiaghayi, M., & Karp, R. M. (2018). Massively parallel symmetry breaking on sparse graphs: MIS and maximal matching. arXiv preprint arXiv:1807.06701.

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@article{behnezhad2018massively,
 title={Massively parallel symmetry breaking on sparse graphs: MIS and maximal matching},
 author={Behnezhad, Soheil and Derakhshan, Mahsa and Hajiaghayi, MohammadTaghi and Karp, Richard M},
 journal={arXiv preprint arXiv:1807.06701},
 year={2018}
}

7/ [PODC'19] Chang, Y. J., Fischer, M., Ghaffari, M., Uitto, J., & Zheng, Y. (2019, July). The complexity of $$(\Delta+ 1)$$ coloring in congested clique, massively parallel computation, and centralized local computation. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing (pp. 471-480).

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@inproceedings{chang2019complexity,
 title={The complexity of ($\Delta$+ 1) coloring in congested clique, massively parallel computation, and centralized local computation},
 author={Chang, Yi-Jun and Fischer, Manuela and Ghaffari, Mohsen and Uitto, Jara and Zheng, Yufan},
 booktitle={Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing},
 pages={471--480},
 year={2019}
}

8/ [FOCS'19] Ghaffari, M., Kuhn, F., & Uitto, J. (2019, November). Conditional hardness results for massively parallel computation from distributed lower bounds. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) (pp. 1650-1663). IEEE.

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@inproceedings{ghaffari2019conditional,
 title={Conditional hardness results for massively parallel computation from distributed lower bounds},
 author={Ghaffari, Mohsen and Kuhn, Fabian and Uitto, Jara},
 booktitle={2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)},
 pages={1650--1663},
 year={2019},
 organization={IEEE}
}

9/ [SODA'20] Ghaffari, M., Nowicki, K., & Thorup, M. (2020). Faster algorithms for edge connectivity via random 2-out contractions. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 1260-1279). Society for Industrial and Applied Mathematics.

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@inproceedings{ghaffari2020faster,
 title={Faster algorithms for edge connectivity via random 2-out contractions},
 author={Ghaffari, Mohsen and Nowicki, Krzysztof and Thorup, Mikkel},
 booktitle={Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms},
 pages={1260--1279},
 year={2020},
 organization={SIAM}
}

Combinatorics on Nam Le

Metric $k$-center

Mathematics - Combinatorics

List of Selected Papers on Algorithms for Large-Scale Graph Processing.