A QPTAS for Facility Location on Unit Disk graphs

TL;DR

This work establishes the first QPTAS for the Uncapacitated Facility Location problem on Unit Disk Graphs, achieving a (1+ε)-approximation in time n^{O_ε(log n)}. The core idea is to adapt planar-graph PTAS techniques to dense UDGs by combining a low-diameter decomposition (δ-chopping via Lee’s results) with a balanced-separator-based dissection tailored to UDGs (Yan 2012). A hierarchical, portal-based dynamic program over a Tarjan-style separator decomposition yields near-optimal solutions on independently bounded-diameter subinstances, with additive and multiplicative errors controlled to sum to O(ε·opt). The paper also provides a PTAS for facility location in bounded regions (constant-size area), leveraging ε-nets and per-cell Prize-Collecting variants, thereby strengthening the practical relevance for dense geometric graphs. Overall, the results bridge a gap between geometric-graph facility location techniques and dense unit-disk structures, offering a powerful framework for extending PTAS/QPTAS methods to UDGs and related graph classes.

Abstract

We study the classic \textsc{(Uncapacitated) Facility Location} problem on Unit Disk Graphs (UDGs). For a given point set $P$ in the plane, the unit disk graph UDG(P) on $P$ has vertex set $P$ and an edge between two distinct points $p, q \in P$ if and only if their Euclidean distance $|pq|$ is at most 1. The weight of the edge $pq$ is equal to their distance $|pq|$. An instance of \fl on UDG(P) consists of a set $C\subseteq P$ of clients and a set $F\subseteq P$ of facilities, each having an opening cost $f_i$. The goal is to pick a subset $F'\subseteq F$ to open while minimizing $\sum_{i\in F'} f_i + \sum_{v\in C} d(v,F')$, where $d(v,F')$ is the distance of $v$ to nearest facility in $F'$ through UDG(P). In this paper, we present the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem. While approximation schemes are well-established for facility location problems on sparse geometric graphs (such as planar graphs), there is a lack of such results for dense graphs. Specifically, prior to this study, to the best of our knowledge, there was no approximation scheme for any facility location problem on UDGs in the general setting.

Theorems & Definitions (23)

• Theorem 1
• Definition 1: $\delta$-chopping operation
• Theorem 2: Lee17
• Theorem 3
• Lemma 1: Corollary 5 cohen2019polynomial
• Lemma 2: Corollary 7 cohen2019polynomial
• Definition 2: Structured Instance with Bounded Aspect Ratio
• Lemma 3: Lemmas 10 and 11 cohen2019polynomial
• Theorem 4: YAN2012
• Theorem 5