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) $\alpha$ approximation if it outputs a solution $S\subseteq F$ satisfying $open(S) + d(S) \leq open(S^*) + \alpha d(S^*)$ 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:无电设施位置问题(UFL)是最基本的组群问题之一:鉴于一组客户(C)美元和一组设施(F)美元(F)美元(F)美元(F)美元(F)美元(F)美元(F)美元(F)美元(R)美元(美元),目标是寻找一套设施(S)美元(F)美元(F)美元(F)美元(F)美元(F)美元(F)美元(F),以尽量减少开机成本(S)和连接费用(d)美元(S):=========== ⁇ p=in C}\min ⁇ c=in s}(p) d)美元(c)美元(美元)。对于UFLL的算法(C)成本(LMP),美元(美元)的算法(FIFIL)值(美元)的算法(SUFI(S)的算法(SO(SO) ), 美元(SUFI(S-S) 美元(SI(S) leveloplevelop levelop legal) ral-ral-ral res(S) legal leg) leg) legleglegal leglegal res(S) legal legal leg) legal leg) legal leg) ma(S) leglegal legal le a结果(我们的算法(S) leglegle) legal le) legal ma(Sil legleglegleg) 和 le) ral)。