Alamdari and Shmoys introduced the following variant of the k -median problem. In this variant, we are given an instance of the k -median problem and a threshold value. Then this variant is the same as the k -median problem except that if the distance between a client i and a facility j is more than the threshold value, then i is not allowed to be connected to j . In this paper, we consider a matroid generalization of this variant of the k -median problem. First, we introduce a generalization of this variant in which the constraint on the number of opened facilities is replaced by a matroid constraint. Then we propose a polynomial-time bicriteria approximation algorithm for this problem by combining the algorithm of Alamdari and Shmoys and the algorithm of Krishnaswamy, Kumar, Nagarajan, Sabharwal, and Saha for a matroid generalization of the k -median problem.

中文翻译：

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距离约束拟阵中值问题

Alamdari 和 Shmoys 介绍了 k 中值问题的以下变体。在这个变体中，我们给出了 k 中值问题的一个实例和一个阈值。那么这个变体与 k 中值问题相同，只是如果客户端 i 和设施 j 之间的距离大于阈值，则不允许 i 连接到 j 。在本文中，我们考虑 k 中值问题的这种变体的拟阵推广。首先，我们介绍了这个变体的推广，其中对开放设施数量的约束被一个拟阵约束代替。然后我们结合 Alamdari 和 Shmoys 的算法以及 Krishnaswamy、Kumar、Nagarajan、Sabharwal 的算法，针对该问题提出了多项式时间双准则逼近算法，