The red-blue median problem considers a set of red facilities, a set of blue facilities, and a set of clients located in some metric space. The goal is to open $k_r$ red facilities and $k_b$ blue facilities such that the sum of the distance from each client to its nearest opened facility is minimized, where $k_r$, $k_b\ge 0$ are two given integers. Designing approximation algorithms for this problem remains an active area of research due to its applications in various fields. However, in many applications, the existence of noisy data poses a big challenge for the problem. In this paper, we consider the prize-collecting red-blue median problem, where the noisy data can be removed by paying a penalty cost. The current best approximation guarantee for the prize-collecting red-blue median problem is a ratio of 24, which was obtained by LP-rounding. We deal with this problem using a local search algorithm. We construct a layered structure of the swap pairs, which yields a $(9+ϵ)$-approximation for the prize-collecting red-blue median problem. Our techniques generalize to a more general prize-collecting τ-color median problem, where the facilities have τ different types, and give a $(4\tau +1+ϵ)$-approximation for the case where τ is a constant.

在红蓝位问题考虑了一套红色设施，一套蓝色的设施，以及一组位于某个度量空间的客户端。目标是开${克}_r$ 红色设施和 ${克}_{乙}$ 蓝色设施，使得从每个客户到最近的开放设施的距离总和最小，其中 ${克}_r$, ${克}_{乙}\ge 0$是两个给定的整数。由于其在各个领域的应用，为此问题设计近似算法仍然是一个活跃的研究领域。然而，在许多应用中，噪声数据的存在给问题带来了很大的挑战。在本文中，我们考虑了奖品收集红蓝中值问题，其中可以通过支付惩罚成本来去除噪声数据。当前对红蓝中位数问题的最佳近似保证是比率为 24，这是通过 LP 舍入获得的。我们使用局部搜索算法来处理这个问题。我们构建了一个交换对的分层结构，它产生了一个$(9+\epsilon )$- 奖品收集红蓝中值问题的近似。我们的技术推广到更一般的奖品收集 τ 颜色中值问题，其中设施有τ种不同类型，并给出$(4\tau +1+\epsilon )$- τ为常数时的近似值。