The Uncapacitated Facility Location (UFL) problem is one of the most fundamental clustering problems: Given a set of clients $C$ and a set of facilities $F$ in a metric space $(C \cup F, dist)$ with facility costs $open : F \to \mathbb{R}^+$, the goal is to find a set of facilities $S \subseteq F$ to minimize the sum of the opening cost $open(S)$ and the connection cost $d(S) := \sum_{p \in C} \min_{c \in S} dist(p, c)$. An algorithm for UFL is called a Lagrangian Multiplier Preserving (LMP) $α$ approximation if it outputs a solution $S\subseteq F$ satisfying $open(S) + d(S) \leq open(S^*) + αd(S^*)$ for any $S^* \subseteq F$. The best-known LMP approximation ratio for UFL is at most $2$ by the JMS algorithm of Jain, Mahdian, and Saberi based on the Dual-Fitting technique. We present a (slightly) improved LMP approximation algorithm for UFL. This is achieved by combining the Dual-Fitting technique with Local Search, another popular technique to address clustering problems. From a conceptual viewpoint, our result gives a theoretical evidence that local search can be enhanced so as to avoid bad local optima by choosing the initial feasible solution with LP-based techniques. Using the framework of bipoint solutions, our result directly implies a (slightly) improved approximation for the $k$-Median problem from 2.6742 to 2.67059.

翻译：无容量设施选址（UFL）问题是最基础的聚类问题之一：给定度量空间$(C \cup F, dist)$中的客户集$C$与设施集$F$，以及设施开启成本函数$open : F \to \mathbb{R}^+$，其目标是寻找设施子集$S \subseteq F$以最小化开启成本$open(S)$与连接成本$d(S) := \sum_{p \in C} \min_{c \in S} dist(p, c)$之和。若算法对任意$S^* \subseteq F$输出的解$S\subseteq F$满足$open(S) + d(S) \leq open(S^*) + αd(S^*)$，则称该算法为拉格朗日乘子保持（LMP）$α$近似算法。基于对偶拟合技术的Jain、Mahdian与Saberi提出的JMS算法，目前实现了最优的UFL问题LMP近似比上界2。本文提出一种（略微）改进的UFL问题LMP近似算法，其核心在于将对偶拟合技术与处理聚类问题的另一主流技术——局部搜索法相结合。从概念层面看，本研究为以下观点提供了理论依据：通过采用基于线性规划的初始可行解，可以增强局部搜索算法以避免陷入不良局部最优解。借助双点解框架，本研究成果直接将$k$-中值问题的近似比从2.6742提升至2.67059。