Chromatic $k$ -median is a frequently encountered problem in the determination of the topological structures of chromosomes. This problem considers a set $\mathcal {C}$ of colored clients and a set $\mathcal {F}$ of facilities located in a metric space, where $|\mathcal {C}\cup \mathcal {F}|=n$ . The goal is to open $k$ facilities and assign each client to an opened facility, such that clients with the same color are assigned to different facilities and the sum of the distance from each client to the corresponding facility is minimized. It was known that the chromatic $k$ -median problem is W[2]-hard if parameterized by $k$ . This rules out the probability of obtaining an exact FPT( $k$ )-time algorithm for the problem. In this paper, we give an FPT( $k$ )-time approximation algorithm for chromatic $k$ -median. The algorithm achieves a $(3+\epsilon)$ -approximation and runs in $(k\epsilon ^{-1})^{O(k)}n^{O(1)}$ time. We propose a different random sampling algorithm for opening facilities, which is the crucial step in getting the constant factor parameterized approximation.

中文翻译：

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色k-中值问题的参数化逼近算法

色度的 $ k $ -中位数是确定染色体拓扑结构时经常遇到的问题。这个问题考虑一套 $ \数学{C} $ 有色客户和一套 $ \数学{F} $ 在公制空间中的设施数量， $ | \ mathcal {C} \ cup \ mathcal {F} | = n $ 。目标是开放 $ k $ 设施，并将每个客户分配给一个开放的设施，以便将具有相同颜色的客户分配给不同的设施，并使每个客户到相应设施的距离之和最小。众所周知， $ k $ -中位数的问题是W [2]-如果参数化是困难的 $ k $ 。这排除了获得精确FPT（ $ k $ ）-问题的时间算法。在本文中，我们给出了FPT（ $ k $ 色的时间近似算法 $ k $ -中位数。该算法实现了 $（3+ \ epsilon）$ -近似并在 $（k \ epsilon ^ {-1}）^ {O（k）} n ^ {O（1）} $ 时间。我们针对开放设施提出了一种不同的随机抽样算法，这是获得恒定因子参数化逼近的关键步骤。