具有双侧边界和野值的容量约束中心问题
Capacitated Center Problems with Two-Sided Bounds and Outliers
摘要：
近年来，容量限制中心问题引起了人们的广泛研究兴趣。给定一组顶点$V$，我们希望找到顶点的子集$S$，称为中心，使得最大聚类半径最小化。此外，$S$中的每个中心应该满足某种容量约束，该容量约束可以是其可以服务的顶点数量的上限或下限。具有单侧界（上界或下界）的容量约束$k$ -中心问题在以前的工作中得到了很好的研究，并得到了一个常数因子近似.我们是第一个研究容量限制中心问题的能力下限和上限（有或没有离群值）。我们假设每个顶点有一个一致的下界和一个非一致的上界。对于开放的情况下，正是$k$中心，我们注意到，最近的LP方法的推广可以实现常数因子近似算法为我们的问题。我们的主要贡献是一个简单的组合算法的情况下，没有基数约束的开放中心的数量。我们的组合算法是简单的，实现更好的常数逼近因子相比，LP方法。
英文摘要：
In recent years, the capacitated center problems have attracted a lot of research interest. Given a set of vertices $V$, we want to find a subset of vertices $S$, called centers, such that the maximum cluster radius is minimized. Moreover, each center in $S$ should satisfy some capacity constraint, which could be an upper or lower bound on the number of vertices it can serve. Capacitated $k$-center problems with one-sided bounds (upper or lower) have been well studied in previous work, and a constant factor approximation was obtained. We are the first to study the capacitated center problem with both capacity lower and upper bounds (with or without outliers). We assume each vertex has a uniform lower bound and a non-uniform upper bound. For the case of opening exactly $k$ centers, we note that a generalization of a recent LP approach can achieve constant factor approximation algorithms for our problems. Our main contribution is a simple combinatorial algorithm for the case where there is no cardinality constraint on the number of open centers. Our combinatorial algorithm is simpler and achieves better constant approximation factor compared to the LP approach.
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