Facility Location Mechanism Design -- Breaking The Deterministic Barrier

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 $\mathbb{R}<sup>2$, we give a random strategyproof mechanism (RR-CWM) achieving an expected approximation ratio of $\frac{4}π \approx 1.27$, 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 $\mathbb{R}<sup>2$. For $\mathbb{R}<sup>d$, 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 $\mathbb{R}<sup>2$, 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 $\mathbb{R}<sup>2$ is $\frac{4}π$ (matching the upper bound of RR-CWM, which is not a GRD mechanism). For $\mathbb{R}<sup>d$, we show a lower bound of $\sqrt{2} - O(1/d)$.

• The paper introduces RR-CWM, which randomly rotates reported locations before applying the coordinate-wise median and achieves the exact 4/π ≈ 1.273 expected approximation ratio in R², improving on the deterministic √2 barrier while remaining universally strategyproof.
• The paper extends the analysis to higher dimensions, proving a ratio between √2 − O(1/√d) and approximately 1.547, and uses concentration, rotation geometry, and Gaussian approximation tools to establish the lower bound.
• The paper shows that random rotation improves learning-augmented facility location and outperforms generalized random-dictator mechanisms in relevant settings, while leaving the optimal randomized ratio and exact high-dimensional guarantee open.

Problem setting and motivation

The paper studies the canonical utilitarian facility location problem in mechanism design: $n$ strategic agents report locations in $\mathbb{R}^d$, a mechanism outputs a single facility location, each agent's cost is the Euclidean distance to that location, and the objective is to minimize the social cost $\mathrm{SC}(P,m)=\sum_i \|p_i - m\|_2$. The central quantity is the expected approximation ratio $\alpha(M)$ of a (possibly randomized) strategyproof mechanism $M$, defined as the worst case over instances of the ratio between expected mechanism cost and optimal (geometric median) cost. Strategyproofness is required in the strong universal sense: truthful reporting is a dominant strategy for every realization of the mechanism's randomness.

On the real line, the median is both strategyproof and optimal. In $\mathbb{R}^d$, $d \ge 2$, the geometric median is manipulable, and the coordinate-wise median (CWM) — median in each coordinate independently — is strategyproof and computationally efficient, but its worst-case ratio is $\sqrt{2}$ in $\mathbb{R}^2$, which is optimal among all deterministic strategyproof mechanisms [goel2023optimality]. In $\mathbb{R}^d$, CWM's ratio is at most $\mathbb{R}^d$0 [gravin2025approximation]. The open question the paper resolves, posed explicitly by Meir (2019) and Goel et al. (2023), is whether randomized mechanisms can strictly beat the deterministic barrier of $\mathbb{R}^d$1.

The RR-CWM mechanism and the exact $\mathbb{R}^d$2 guarantee

The proposed mechanism, RR-CWM (Randomly Rotated Coordinate-wise Median), samples a Haar-uniform rotation matrix $\mathbb{R}^d$3, rotates all reported points by $\mathbb{R}^d$4, applies CWM to the rotated dataset, and rotates the result back by $\mathbb{R}^d$5. Because agents have no influence over the sampled rotation, universal strategyproofness is inherited from CWM, and the mechanism runs in $\mathbb{R}^d$6 time — linear in $\mathbb{R}^d$7 for fixed $\mathbb{R}^d$8.

The key structural insight is that the deterministic $\mathbb{R}^d$9 barrier stems from an axis-alignment mismatch: CWM minimizes the $\mathrm{SC}(P,m)=\sum_i \|p_i - m\|_2$0 cost in fixed coordinates, while the objective is the rotationally invariant $\mathrm{SC}(P,m)=\sum_i \|p_i - m\|_2$1 norm. A uniform random rotation replaces the worst-case $\mathrm{SC}(P,m)=\sum_i \|p_i - m\|_2$2/$\mathrm{SC}(P,m)=\sum_i \|p_i - m\|_2$3 distortion with its average distortion. The bridge is an elementary but sharp lemma: for any fixed $\mathrm{SC}(P,m)=\sum_i \|p_i - m\|_2$4 and uniformly random rotation $\mathrm{SC}(P,m)=\sum_i \|p_i - m\|_2$5, $\mathrm{SC}(P,m)=\sum_i \|p_i - m\|_2$6. Combining this with (i) rotational isometry of $\mathrm{SC}(P,m)=\sum_i \|p_i - m\|_2$7 distances, (ii) $\mathrm{SC}(P,m)=\sum_i \|p_i - m\|_2$8, and (iii) the fact that each coordinate median minimizes the projected $\mathrm{SC}(P,m)=\sum_i \|p_i - m\|_2$9 cost yields the main upper bound.

The main result: in $\alpha(M)$0, the expected approximation ratio of RR-CWM is exactly $\alpha(M)$1. This is a tight two-sided result: the matching lower bound is established via a "two clusters and an outlier" instance — $\alpha(M)$2 points at $\alpha(M)$3, $\alpha(M)$4 at $\alpha(M)$5, and one at $\alpha(M)$6 with $\alpha(M)$7 — for which the rotated coordinate-wise median is pulled by the outlier for all but an $\alpha(M)$8-measure set of angles, forcing expected cost at least $\alpha(M)$9 against an optimum of $M$0.

Since $M$1, this closes the long-standing open problem of separating randomized and deterministic strategyproof mechanisms for single-facility utilitarian location in $M$2. The separation is strict and quantitative: a $M$3 improvement in approximation ratio, achieved with a black-box randomization that preserves universal truthfulness. The upper bound is also generalized: for any $M$4 cost with $M$5, RR-CWM achieves ratio $M$6 in the plane.

Higher dimensions

For $M$7, the paper proves

$M$8

The upper bound follows trivially from the worst-case bound on CWM, since randomization cannot hurt in expectation. The lower bound reuses the two-clusters-plus-outlier construction in $M$9 (with $\mathbb{R}^d$0 clusters on $\mathbb{R}^d$1 and an outlier at $\mathbb{R}^d$2), but the analysis requires substantially heavier machinery: the Diaconis–Freedman theorem is used to argue that scaled coordinates of Haar-uniform rotations are close in total variation to i.i.d. Gaussians, showing the "bad event" (the outlier capturing a coordinate median) contributes only $\mathbb{R}^d$3 mass; concentration of measure for Lipschitz functions on $\mathbb{R}^d$4 (with an $\mathbb{R}^d$5-Lipschitz constant established for the cost function) then converts a second-moment bound into a first-moment bound. The result is that each cluster point sits at distance $\mathbb{R}^d$6 from the returned facility, giving $\mathbb{R}^d$7 against $\mathbb{R}^d$8.

The paper conjectures that $\mathbb{R}^d$9 (up to $d \ge 2$0) is the true ratio of RR-CWM in all dimensions; the gap between $d \ge 2$1 and $d \ge 2$2 remains open.

Learning-augmented improvements

The random rotation technique transfers directly to learning-augmented settings, yielding strict improvements in both standard prediction models.

Output prediction model [agrawal2022learning]: the CMP mechanism augments the dataset with $d \ge 2$3 copies of the predicted optimal location $d \ge 2$4 and applies CWM. Replacing CWM with RR-CWM yields the RR-CMP mechanism, which inherits CMP's consistency guarantees and improves robustness. The mechanism is $d \ge 2$5-consistent and $d \ge 2$6-robust, with a smooth interpolation in the prediction error $d \ge 2$7; specifically, its ratio is bounded by $d \ge 2$8. The robustness improvement relies on a robust statistics lemma of independent interest: in any metric space, the 1-median computed on a dataset corrupted by a $d \ge 2$9-fraction of inserted points achieves cost at most $\sqrt{2}$0 times the cost of the clean median — an insertion-only corruption analysis that is sharper than the edit-corruption bounds of Barak et al. (2024). A calculation shows strict improvement over CMP whenever $\sqrt{2}$1 (for any $\sqrt{2}$2), or $\sqrt{2}$3 with sufficiently large $\sqrt{2}$4. Because Agrawal et al. proved CMP's trade-off optimal among deterministic anonymous strategyproof mechanisms, this result extends the deterministic/randomized separation to the learning-augmented setting.

MAC input prediction model [barak2024mac]: replacing CWM with RR-CWM in the "predictions-only or CWM" template improves the worst-case fallback arm from $\sqrt{2}$5 to $\sqrt{2}$6, giving an approximation ratio of $\sqrt{2}$7 when a $\sqrt{2}$8 fraction of agent-location predictions are correct.

Limitations of generalized random dictators

The paper establishes lower bounds for GRD (Generalized Random Dictator) mechanisms — any mechanism that outputs only agent-reported locations, a class encompassing Uniform Random Dictator, Proportional, Inverse-Proportional, Phantom, Random Rank, PCD, and percentile-style mechanisms.

• $\sqrt{2}$9: any GRD mechanism has expected ratio at least $\mathbb{R}^2$0. The proof uses $\mathbb{R}^2$1 evenly spaced points on the unit circle; the optimum places the facility at the origin with cost $\mathbb{R}^2$2, while any reported point incurs cost $\mathbb{R}^2$3 via the trigonometric identity $\mathbb{R}^2$4.
• $\mathbb{R}^2$5: any GRD mechanism has expected ratio at least $\mathbb{R}^2$6, via a probabilistic-method argument on i.i.d. uniform points on the unit sphere, using a binomial-expansion bound on the expected distance between random sphere vectors and Hoeffding concentration.

These bounds carry two implications. First, RR-CWM — which is not a GRD mechanism, as it may return a non-reported point — is (weakly) better than every GRD mechanism in $\mathbb{R}^2$7. Second, any future improvement beyond $\mathbb{R}^2$8 in $\mathbb{R}^2$9 requires mechanisms that output locations off the input point set; the GRD design template is provably insufficient.

Relation to the projection median

The paper clarifies the connection to the projection median [durocher2009projection], which equals the average of RR-CWM over all rotations (an identity via the Haar measure on $\mathbb{R}^d$0). Jensen's inequality implies the projection median's approximation ratio is at most RR-CWM's expected ratio, so the paper's $\mathbb{R}^d$1 upper bound immediately improves the projection median's known bound from $\mathbb{R}^d$2 [basu2012projection] to a constant $\mathbb{R}^d$3. However, the two quantities are not identical — a three-point example shows the projection median can achieve strictly lower cost than the RR-CWM expectation — and the projection median is not strategyproof (its ratio in $\mathbb{R}^d$4 is strictly below the deterministic lower bound of $\mathbb{R}^d$5, which is impossible for a strategyproof mechanism). Closing the projection median's own gap, $\mathbb{R}^d$6 in $\mathbb{R}^d$7, cannot be achieved through RR-CWM analysis alone and remains open.

Limitations and open questions

Several caveats qualify the results. The exact $\mathbb{R}^d$8 guarantee is specific to $\mathbb{R}^d$9; in $\mathbb{R}^d$00 the analysis leaves a gap of roughly $\mathbb{R}^d$01, and the conjecture that RR-CWM achieves $\mathbb{R}^d$02 for all $\mathbb{R}^d$03 is unproven — notably, the direct $\mathbb{R}^d$04-bridge technique from the planar proof degrades to $\mathbb{R}^d$05 in high dimensions, so the $\mathbb{R}^d$06 lower bound required different tools, and an upper bound of comparable sharpness may require yet others. Whether the $\mathbb{R}^d$07 ratio is optimal among all randomized strategyproof mechanisms in $\mathbb{R}^d$08 is open; the GRD lower bound does not preclude better non-GRD mechanisms. The learning-augmented improvement over CMP holds only for restricted ranges of the trust parameter $\mathbb{R}^d$09 and error $\mathbb{R}^d$10 (strictly, $\mathbb{R}^d$11, or larger $\mathbb{R}^d$12 with sufficiently large error), and CMP remains superior for small $\mathbb{R}^d$13 with near-perfect predictions. The paper does not address the egalitarian (maximum cost) objective, where the deterministic barrier is $\mathbb{R}^d$14 and whether randomization helps is unknown. Finally, the $\mathbb{R}^d$15 lower bound for RR-CWM is asymptotic and instance-specific; pinning down the exact worst-case profile ratio, as suggested by Goel et al., remains unresolved.

Conclusion

This paper resolves a sixteen-year-old open question by showing that a uniform random rotation applied before the coordinate-wise median yields a universally strategyproof, linear-time mechanism with expected approximation ratio exactly $\mathbb{R}^d$16 in $\mathbb{R}^d$17, strictly breaking the tight $\mathbb{R}^d$18 deterministic barrier. The same technique improves consistency–robustness trade-offs in learning-augmented facility location beyond the provably optimal deterministic frontier, and matching lower bounds for GRD mechanisms establish that output-restricted randomization cannot compete. The results position random rotation as a general design primitive for Euclidean mechanism design, while leaving the exact optimal randomized ratio — in $\mathbb{R}^d$19 and in higher dimensions — as the natural next target.