Breaching the 2 LMP Approximation Barrier for Facility Location with Applications to k-Median

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 ∪ F, dist) with facility costs open : F → R⁺, the goal is to find a set of facilities S ⊆ F to minimize the sum of the opening cost $open(S)$ and the connection cost d(S) := ∑p ∈ C minc ∈ S dist(p, c). An algorithm for UFL is called a Lagrangian Multiplier Preserving (LMP) $α$ approximation if it outputs a solution S⊆ F satisfying open(S) + d(S) ≤ open(S^*) + αd(S^*) for any S^* ⊆ 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.