Ordered k-Median with Outliers

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.