MAC Advice for Facility Location Mechanism Design

TL;DR

This work studies facility location under Mostly Approximately Correct (MAC) predictions, where a prediction for every agent's location can be adversarially wrong on a small $\delta$-fraction while the rest are accurate within $\varepsilon$. It develops robustness tools for location estimators, notably quantifying the $1$-median's continuous robustness under corruptions and proposing a balanced-$k$-median variant that remains robust for $k\ge 2$. Leveraging these tools, the authors design both deterministic and randomized mechanisms that improve over no-prediction baselines: a deterministic single-facility mechanism in $\mathbb{R}^d$ with ratio $\min\{1+\frac{4\delta}{1-2\delta},\sqrt{d}\}$, and a randomized $2$-facility-on-a-line mechanism with expected ratio $3.6+O(\delta)$, along with a deterministic balanced-$k$-facility mechanism with ratio $1+\frac{4k}{b-2-2k}$ for constant $k$ and $\beta$-balanced clusters. The work also introduces the Second Facility Location problem and the Big-Cluster-Center estimator to obtain robust first-facility estimates and integrates these ideas to yield an effective mechanism design framework that uses MAC predictions without succumbing to worst-case errors. Overall, the paper demonstrates that MAC predictions, when paired with agent reports and robust estimators, can surpass traditional no-prediction bounds in facility location and invites further exploration of robust prediction-driven mechanisms in broader domains.

Abstract

Algorithms with predictions have attracted much attention in the last years across various domains, including variants of facility location, as a way to surpass traditional worst-case analyses. We study the $k$-facility location mechanism design problem, where the $n$ agents are strategic and might misreport their location. Unlike previous models, where predictions are for the $k$ optimal facility locations, we receive $n$ predictions for the locations of each of the agents. However, these predictions are only "mostly" and "approximately" correct (or MAC for short) -- i.e., some $δ$-fraction of the predicted locations are allowed to be arbitrarily incorrect, and the remainder of the predictions are allowed to be correct up to an $\varepsilon$-error. We make no assumption on the independence of the errors. Can such predictions allow us to beat the current best bounds for strategyproof facility location? We show that the $1$-median (geometric median) of a set of points is naturally robust under corruptions, which leads to an algorithm for single-facility location with MAC predictions. We extend the robustness result to a "balanced" variant of the $k$ facilities case. Without balancedness, we show that robustness completely breaks down, even for the setting of $k=2$ facilities on a line. For this "unbalanced" setting, we devise a truthful random mechanism that outperforms the best known result of Lu et al. [2010], which does not use predictions. En route, we introduce the problem of "second" facility location (when the first facility's location is already fixed). Our findings on the robustness of the $1$-median and more generally $k$-medians may be of independent interest, as quantitative versions of classic breakdown-point results in robust statistics.

Figures (5)

• Figure 1: Illustration of the problem settings. The black points are the real agent locations. The yellow points are the locations reported by the agents and the blue points are the predicted locations. For a strategyproof mechanism, the yellow points and the black points will overlap.
• Figure 2: Illustration of case (4) where $m' \ge g_L$. On the top we have $h_L, h_R$, computed on the "predicted" locations $X'$. On the bottom we have the $g_L, g_R$, the $\mathop{\mathrm{2-Medians}}\nolimits$ of the "real" locations $X$.
• Figure 3: Illustration of the 4 cases. On the top of each case drawing we have the "estimated"/"predicted" locations $H$, computed on $X'$. On the bottom we have the "real" locations, $G$, computed on $X$.
• Figure 4: Illustration of case (4) where $m' \ge g_L$. On the top we have $h_L, h_R$, computed on the "predicted" locations $X'$. On the bottom we have the $g_L, g_R$, the $\mathop{\mathrm{2-Medians}}\nolimits$ of the "real" locations $X$. $L'$ and $R'$ are the disjoint partitions of $X'$ into two multi-sets of points: those closer to $h_L$ and those closer to $h_R$ (respectively). Similarly $L$ and $R$ are the disjoint multi-sets of $g_L, g_R$. ${L'}_l, {L'}_r$ are the disjoint partitions of $L'$ into two multi-sets: all points to the left of $h_L$ and all points to the right of $h_R$. In a similar manner $R', L, R$ are partitioned into their left and right parts (${R'}_l, {R'}_r, L_l, L_r, R_l, R_r$). We can also see $S, T, U ,V$ which is the disjoint partition of $L$ determined by the points $h_L, m', h_R$. Finally, we have $U_l$ and $U_r$ which are the left and right parts of $U$.
• Figure 5: Illustration of case (4) where $m' < g_L$. On the top we have the "estimated" locations $h_L, h_R$, computed on $X'$. On the bottom we have the "real" locations, $g_L, g_R$, computed on $X$. $L', R', {L'}_l, {L'}_r, {R'}_l, {R'}_r, S, T, U ,V, U_l, U_r$ are all the same as before. The difference is that $m'$ is now to the left of $g_L$.

Theorems & Definitions (65)

• Example 1: $2$-facility location sensitivity
• Definition 1
• Definition 2
• Definition 3: Mostly Approximately Correct (Mac) Predictions
• Definition 4: k-medians cost function
• Definition 5: Location Estimator
• Definition 6
• Definition 7: Distance Robustness
• Definition 8: Approximation Robustness
• Lemma 2