Approximating k-hop minimum spanning trees in Euclidean metrics
Information Processing Letters
The Kinetic Facility Location Problem
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
Cluster-Swap: A Distributed K-median Algorithm for Sensor Networks
WI-IAT '09 Proceedings of the 2009 IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology - Volume 02
Linear-time approximation schemes for clustering problems in any dimensions
Journal of the ACM (JACM)
A quasi-polynomial time approximation scheme for Euclidean capacitated vehicle routing
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Budgeted red-blue median and its generalizations
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Some results on approximate 1-median selection in metric spaces
Theoretical Computer Science
The traveling salesman problem: low-dimensionality implies a polynomial time approximation scheme
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Fast k-clustering queries on embeddings of road networks
Proceedings of the 3rd International Conference on Computing for Geospatial Research and Applications
Deterministic sublinear-time approximations for metric 1-median selection
Information Processing Letters
The euclidean k-supplier problem
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
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This paper provides a randomized approximation scheme for the $k$-median problem when the input points lie in the $d$-dimensional Euclidean space. The worst-case running time is $O(2^{O((\log(1/\epsilon) / \varepsilon)^{d-1})} n \log^{d+6} n ),$ which is nearly linear for any fixed $\varepsilon$ and $d$. Moreover, our method provides the first polynomial-time approximation scheme for and uncapacitated facility location instances in $d$-dimensional Euclidean space for any fixed $d 2.$ Our work extends techniques introduced originally by Arora for the Euclidean traveling salesman problem (TSP). To obtain the improvement we develop a structure theorem to describe hierarchical decomposition of solutions. The theorem is based on an adaptive decomposition scheme, which guesses at every level of the hierarchy the structure of the optimal solution and accordingly modifies the parameters of the decomposition. We believe that our methodology is of independent interest and may find applications to further geometric problems.