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) := Σp∈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 [STOC'02, J.ACM'03] based on the Dual-Fitting technique. The lack of progress on improving the upper bound on αLMP in the last two decades raised the natural question whether αLMP = 2.We answer this question negatively by presenting 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. In more detail, we use the LMP solution S produced by JMS to seed a local search algorithm. We show that local search substantially improves S unless a big fraction of the connection cost of S is associated with facilities of relatively small opening costs. In the latter case however the analysis of Jain, Mahdian, and Saberi can be improved (i.e., S is cheaper than expected). To summarize: Either S is close enough to the optimum, or it must belong to the local neighborhood of a good enough local optimum. 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.Our result directly implies a (slightly) improved approximation for the related k-Median problem, another fundamental clustering problem: Given (C ∪ F, dist) as in a UFL instance and an integer k ∈ ℕ, find S ⊆ F with |S| = k that minimizes d(S). The current best approximation algorithms for k-Median are based on the following framework: use an LMP α approximation algorithm for UFL to build an α approximate bipoint solution for k-Median, and then round it with a ρBR approximate bipoint rounding algorithm. This implies an α · ρBR approximation. The current-best value of ρBR is 1.338 by Byrka, Pensyl, Rybicki, Srinivasan, and Trinh [SODA'15, TALG'17], which yields 2.6742-approximation. Combining their algorithm with our refined LMP algorithm for UFL (replacing JMS) gives a 2.67059-approximation.* The full version of the paper can be accessed at https://arxiv.org/abs/2207.05150