We study a natural generalization of the celebrated ordered k-median problem, named robust ordered k-median, also known as ordered k-median with outliers. We are given facilities β± and clients π in a metric space (β±βͺπ,d), parameters k,m β β€_+ and a non-increasing non-negative vector w β β_+^m. We seek to open k facilities F β β± and serve m clients C β π, inducing a service cost vector c = {d(j,F):j β C}; the goal is to minimize the ordered objective w^β€c^β, where d(j,F) = min_{i β F}d(j,i) is the minimum distance between client j and facilities in F, and c^β β β_+^m is the non-increasingly sorted version of c. Robust ordered k-median captures many interesting clustering problems recently studied in the literature, e.g., robust k-median, ordered k-median, etc. We obtain the first polynomial-time constant-factor approximation algorithm for robust ordered k-median, achieving an approximation guarantee of 127. The main difficulty comes from the presence of outliers, which already causes an unbounded integrality gap in the natural LP relaxation for robust k-median. This appears to invalidate previous methods in approximating the highly non-linear ordered objective. To overcome this issue, we introduce a novel yet very simple reduction framework that enables linear analysis of the non-linear objective. We also devise the first constant-factor approximations for ordered matroid median and ordered knapsack median using the same framework, and the approximation factors are 19.8 and 41.6, respectively.
@InProceedings{deng_et_al:LIPIcs.APPROX/RANDOM.2022.34, author = {Deng, Shichuan and Zhang, Qianfan}, title = {{Ordered k-Median with Outliers}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {34:1--34:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.34}, URN = {urn:nbn:de:0030-drops-171560}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.34}, annote = {Keywords: clustering, approximation algorithm, design and analysis of algorithms} }
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