Abstract

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 ℱ and clients 𝒞 in a metric space (ℱ∪𝒞,d), parameters k,m ∈ ℤ_+ and a non-increasing non-negative vector w ∈ ℝ_+^m. We seek to open k facilities F ⊆ ℱ and serve m clients C ⊆ 𝒞, inducing a service cost vector c = {d(j,F):j ∈ C}; the goal is to minimize the ordered objective w^⊤c^↓, where d(j,F) = min_{i ∈ F}d(j,i) is the minimum distance between client j and facilities in F, and c^↓ ∈ ℝ_+^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/opus/volltexte/2022/17156},
  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}
}