arXiv 1608.017002016-08cs.DS

具有下界和离群点的聚类问题的近似算法

Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers

Sara Ahmadian, Chaitanya Swamy

摘要：

我们考虑聚类问题与{ \em非一致下界和离群值}，并获得这些问题的{ \em第一近似保证}。我们有一组具有下界$\{L_i\}_{i\in\F}$的设施$\F$和一组位于公共度量空间$\{c(i,j)\}_{i,j\in\F\cup\D}$中的客户端$\D$，以及下界$k$，$m$。可行的解决方案是一对$\bigl(S\sse\F,σ:\D\mapsto S\cup\{\mathsf{out}\}\bigr)$，其中$σ$指定客户端分配，使得$|S|\leq k$、用于所有$i\in S$的$|σ^{-1}(i)|\geq L_i$和$|σ^{-1}(\mathsf{out})|\leq m$。在{ \em下界最小半径和与离群值}（\lbksro）问题中，目标是最小化$\sum_{i\in S}\max_{j\inσ^{-1}(i)}c(i,j)$，而在{ \em下界$k$ -供应商与离群值}（\lbkso）问题中，目标是最小化$\max_{i\in S}\max_{j\inσ^{-1}(i)}c(i,j)$。我们获得了\lbksro的$12.365$的近似因子，其对于非异常值版本（即，$m=0$）。当我们分别考虑下界和离群值时，这些也构成了最小半径和目标的{\em first}近似界。我们应用原始-对偶方法的松弛，我们拉格朗日化的$|S|\leq k$约束。我们的算法的主要技术贡献和新颖性是，从用于这种约束问题的标准范式出发，我们得到一个$O(1)$ -近似{ \em，尽管事实上，我们没有得到一个拉格朗日乘子保持算法的拉格朗日松弛}。我们相信，我们的想法有{更广泛的适用性，其他聚类问题的离群值以及。}我们分别获得了\lbkso及其非离群值版本的$5$和$3$的近似因子。这些是具有{\em非均匀}下限的$k$ -供应商的{ \em第一}近似结果。

英文摘要：

We consider clustering problems with {\em non-uniform lower bounds and outliers}, and obtain the {\em first approximation guarantees} for these problems. We have a set $\F$ of facilities with lower bounds $\{L_i\}_{i\in\F}$ and a set $\D$ of clients located in a common metric space $\{c(i,j)\}_{i,j\in\F\cup\D}$, and bounds $k$, $m$. A feasible solution is a pair $\bigl(S\sse\F,σ:\D\mapsto S\cup\{\mathsf{out}\}\bigr)$, where $σ$ specifies the client assignments, such that $|S|\leq k$, $|σ^{-1}(i)|\geq L_i$ for all $i\in S$, and $|σ^{-1}(\mathsf{out})|\leq m$. In the {\em lower-bounded min-sum-of-radii with outliers} (\lbksro) problem, the objective is to minimize $\sum_{i\in S}\max_{j\inσ^{-1}(i)}c(i,j)$, and in the {\em lower-bounded $k$-supplier with outliers} (\lbkso) problem, the objective is to minimize $\max_{i\in S}\max_{j\inσ^{-1}(i)}c(i,j)$. We obtain an approximation factor of $12.365$ for \lbksro, which improves to $3.83$ for the non-outlier version (i.e., $m=0$). These also constitute the {\em first} approximation bounds for the min-sum-of-radii objective when we consider lower bounds and outliers {\em separately}. We apply the primal-dual method to the relaxation where we Lagrangify the $|S|\leq k$ 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 {\em despite the fact that we do not obtain a Lagrangian-multiplier-preserving 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 \lbkso and its non-outlier version. These are the {\em first} approximation results for $k$-supplier with {\em non-uniform} lower bounds.

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