Improved Approximation Guarantees for Lower-Bounded Facility Location

Abstract

We consider the lower-bounded facility location (LBFL) problem, which is a generalization of uncapacitated facility location (UFL), where each open facility is required to serve a certain minimum amount of demand. The current best approximation ratio for LBFL is 448 [17]. We substantially advance the state-of-the-art for LBFL by improving its approximation ratio from 448 [17] to 82.6.

Our improvement comes from a variety of ideas in algorithm design and analysis, which also yield new insights into LBFL. Our chief algorithmic novelty is to present an improved method for solving a more-structured LBFL instance obtained from $\ensuremath{\mathcal I}$ via a bicriteria approximation algorithm for LBFL, wherein all clients are aggregated at a subset $\ensuremath{\mathcal F}'$ of facilities, each having at least α M co-located clients (for some α ∈ [0,1]). The algorithm in [17] proceeds by reducing $\ensuremath{\mathcal I}_2(\ensuremath{\alpha})$ to CFL. One of our key insights is that one can reduce the resulting LBFL instance, denoted $\ensuremath{\mathcal I}_2(\ensuremath{\alpha})$, to a problem we introduce, called capacity-discounted UFL (CDUFL), which is a structured special case of capacitated facility location (CFL). We give a simple local-search algorithm for CDUFL based on add, delete, and swap moves that achieves a significantly-better approximation ratio than the current-best approximation ratio for CFL, which is one of the reasons behind our algorithm’s improved approximation ratio.

A full version [1] is available on the CS arXiv.