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.