Coordinate-wise median is a tight 2^{1-1/max(p,q)}-approximation for p-norm facility location in ℓ_q(R^2) and at most 3-approx in any dimension for all p,q≥1.
Facility Location Mechanism Design: Breaking The Deterministic Barrier
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abstract
We study the facility location mechanism design problem where $n$ agents report their locations in Euclidean space, and the output is a single facility location. The cost function of each agent is the distance from the returned facility, and the objective is to minimize the social cost function (the sum of agent costs) in a strategyproof way. Our contributions: 1. Breaking the deterministic barrier. For $\mathbb{R}^2$, we give a random strategyproof mechanism (RR-CWM) achieving an expected approximation ratio of $\frac{4}{\pi} \approx 1.27$, which strictly improves upon the best deterministic strategyproof mechanism (which has a $\sqrt{2} \approx 1.41$ ratio). This closes the open problem of separating deterministic and random mechanisms for utilitarian facility location mechanism design in $\mathbb{R}^2$. For $\mathbb{R}^d$, we show that the expected approximation ratio of our mechanism is in $[1.41 - O(1/\sqrt{d}), 1.547]$. 2. Improved learning augmented mechanisms through randomization. We show our ideas can achieve better performance in the learning augmented setting in $\mathbb{R}^2$, where in addition to the input the mechanism also receives predictions. For the output prediction model of Agrawal et al. 2022 we show an improved expected consistency-robustness trade-off. Our results also imply improved performance for the input MAC predictions model of Barak et al. 2024. 3. The limitations of Random Dictators. We show a lower bound for the common mechanism class of GRD (Generalized Random Dictator) mechanisms, where only locations reported by the agents may be returned. We show that any GRD mechanism has a larger expected approximation ratio than our RR-CWM mechanism, as our lower bound for $\mathbb{R}^2$ is $\frac{4}{\pi}$ (matching the upper bound of RR-CWM, which is not a GRD mechanism). For $\mathbb{R}^d$, we show a lower bound of $\sqrt{2} - O(1/d)$.
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cs.GT 2years
2026 2verdicts
ACCEPT 2representative citing papers
Confirms that coordinate-wise median achieves tight approximation ratios of 2^{1-1/p} (p>=2) and sqrt(2) (1<=p<=2) and is optimal among deterministic anonymous strategyproof mechanisms; randomized mechanisms improve for p around 1.6 and above.
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Strategic Facility Location with $p$-Norm Social Costs
Coordinate-wise median is a tight 2^{1-1/max(p,q)}-approximation for p-norm facility location in ℓ_q(R^2) and at most 3-approx in any dimension for all p,q≥1.
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Strategyproof Mechanisms for Euclidean Facility Location Problems under $L_p$-norm Social Cost
Confirms that coordinate-wise median achieves tight approximation ratios of 2^{1-1/p} (p>=2) and sqrt(2) (1<=p<=2) and is optimal among deterministic anonymous strategyproof mechanisms; randomized mechanisms improve for p around 1.6 and above.