eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-09-15
34:1
34:22
10.4230/LIPIcs.APPROX/RANDOM.2022.34
article
Ordered k-Median with Outliers
Deng, Shichuan
1
Zhang, Qianfan
1
IIIS, Tsinghua University, Beijing, China
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol245-approx-random2022/LIPIcs.APPROX-RANDOM.2022.34/LIPIcs.APPROX-RANDOM.2022.34.pdf
clustering
approximation algorithm
design and analysis of algorithms