Facility Location with Public Locations and Private Doubly-Peaked Costs

TL;DR

The paper studies facility location when agent positions are public but their preferred distances to the facility are private, modeling doubly-peaked costs. It provides strong additive lower bounds for both deterministic and randomized truthful mechanisms in 1D (and extends to 2D with $L_1$), while proposing simple and refined mechanisms based on medians. The Median algorithm offers a universal baseline with additive gap $\le 2nB$ (and tighter bounds under near-median conditions), and the deterministic Median-Plus mechanism achieves comparable performance with potential $\Theta(nB)$ improvements on skewed inputs; a 2D $L_1$ extension is also developed. Together, these results illuminate the trade-offs between truthfulness and approximation quality in public-location facility placement with private distance preferences, offering practical strategyproof mechanisms and tight hardness benchmarks.

Abstract

In the facility location problem, the task is to place one or more facilities so as to minimize the sum of the agent costs for accessing their nearest facility. Heretofore, in the strategic version, agent locations have been assumed to be private, while their cost measures have been public and identical. For the most part, the cost measure has been the distance to the nearest facility. However, in multiple natural settings, such as placing a firehouse or a school, this modeling does not appear to be a good fit. For it seems natural that the agent locations would be known, but their costs might be private information. In addition, for these types of settings, agents may well want the nearest facility to be at the right distance: near, but not too near. This is captured by the doubly-peaked cost introduced by Filos-Ratsikas et al. (AAMAS 2017). In this paper, we re-examine the facility location problem from this perspective: known agent locations and private preferred distances to the nearest facility. We then give lower and upper bounds on achievable approximations, focusing on the problem in 1D, and in 2D with an $L_1$ distance measure.

Figures (10)

• Figure 1: Instance $\mathcal{I}_1$. The locations are indicated by the circle, diamond and square respectively, and the corresponding preferred locations are indicated by the round brackets, the angle brackets, and the square brackets.
• Figure 2: Instance $\mathcal{I}_2$
• Figure 3: Two cases when $x_i \geq \operatorname{med}$. The red star represents $f(\mathcal{I})$---the output of Median-Plus when agents report truthfully. The black bracket shows agent $i$'s true preferred facility location. The orange bracket is agent $i$'s apparent preferred facility location when she misreports $b'_i < b_i$. The green bracket shows the location when she misreports $b'_i > b_i$.
• Figure 4: Skewed Instance. Median and one of its preferred location is marked with black color. Agents to the left of median are colored orange along with their preferred locations and agents to the right of median are colored purple. The output of Median-Plus $\operatorname{medp}$ is marked in a red star.
• Figure 5: Instance $\mathcal{I}_2$

Theorems & Definitions (48)

• definition thmcounterdefinition: For facility location mechanisms with doubly-peaked preferences
• lemma thmcounterlemma
• theorem thmcountertheorem
• theorem thmcountertheorem
• proof
• proof
• proof : Proof of \ref{['thm:det-1d-hardness']}
• proof
• lemma thmcounterlemma
• proof