We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {L i } i∈F and a set D of clients located in a common metric space {c(i, j)} i,j∈F ∪D , and bounds k, m. A feasible solution is a pair S ⊆ F, σ : D → S ∪ {out} , where σ specifies the client assignments, such that |S| ≤ k, In the lower-bounded min-sum-of-radii with outliers (LBkSRO) problem, the objective is to minimize i∈S max j∈σ −1 (i) c(i, j), and in the lower-bounded k-supplier with outliers (LBkSupO) problem, the objective is to minimize max i∈S max j∈σ −1 (i) c(i, j). We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the nonoutlier version (i.e., m = 0). These also constitute the first approximation bounds for the min-sum-ofradii objective when we consider lower bounds and outliers separately. We apply the primal-dual method to the relaxation where we Lagrangify the |S| ≤ k constraint. The chief technical contribution and novelty of our algorithm is that, departing from the standard paradigm used for such constrained problems, we obtain an O(1)-approximation despite the fact that we do not obtain a Lagrangian-multiplierpreserving algorithm for the Lagrangian relaxation. We believe that our ideas have broader applicability to other clustering problems with outliers as well. We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds.
Ahmadian and Swamy REF consider the combination of k-supplier with outliers with (non-uniform) lower bounds and derive a 5-approximation.
Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers
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Computer Science Mathematics