Learning-Augmented Metric Distortion via $(p,q)$-Veto Core

TL;DR

The paper studies metric distortion in a setting with candidates and voters in a metric space where only ordinal input is available, and the social cost is cardinal. It introduces the generalized $(p,q)$-veto core and proves universal distortion bounds for algorithms selecting veto-core candidates, recovering the known optimal distortion $3$ in a canonical setting. It then leverages a learning-augmented framework to design a prediction-guided algorithm, LA$_\delta$, achieving the optimal robustness-consistency trade-off, with refined guarantees that depend on prediction quality and voter decisiveness. It also establishes an impossibility result for full-metric predictions and extends the analysis to $\oldsymbol{\alpha}$-decisive voters, providing nuanced distortion bounds and suggesting fruitful directions for future work in veto-core parameterizations and randomized extensions.

Abstract

In the metric distortion problem there is a set of candidates $C$ and voters $V$ in the same metric space. The goal is to select a candidate minimizing the social cost: the sum of distances of the selected candidate from all the voters, and the challenge arises from the algorithm receiving only ordinaL input: each voter's ranking of candidate, while the objective function is cardinal, determined by the underlying metric. The distortion of an algorithm is its worst-case approximation factor of the optimal social cost. A key concept here is the (p,q)-veto core, with $p\in Δ(V)$ and $q\in Δ(C)$ being normalized weight vectors representing voters' veto power and candidates' support, respectively. The (p,q)-veto core corresponds to a set of winners from a specific class of deterministic algorithms. Notably, the optimal distortion of $3$ is obtained from this class, by selecting veto core candidates using uniform $p$ and $q$ proportional to candidates' plurality scores. Bounding the distortion of other algorithms from this class is an open problem. Our contribution is twofold. First, we establish upper bounds on the distortion of candidates from the (p,q)-veto core for arbitrary weight vectors $p$ and $q$. Second, we revisit the metric distortion problem through the \emph{learning-augmented} framework, which equips the algorithm with a (machine-learned) prediction regarding the optimal candidate. The quality of this prediction is unknown, and the goal is to optimize the algorithm's performance under accurate predictions (consistency), while simultaneously providing worst-case guarantees under arbitrarily inaccurate predictions (robustness). We propose an algorithm that chooses candidates from the (p,q)-veto core, using a prediction-guided q vector and, leveraging our distortion bounds, we prove that this algorithm achieves the optimal robustness-consistency trade-off.

Figures (4)

• Figure 1: On the left, a plot showing the consistency and robustness bounds achieved by $\textsc{LA}_\delta$ as a function of the confidence parameter $\delta$. On the right, a plot of the Pareto-frontier capturing the best possible trade-off between robustness and consistency, all of which can be achieved by $\textsc{LA}_\delta$ for the appropriate choice of $\delta$.
• Figure 2: An example of a domination graph and a VDG graph for an instance with preference profile $\sigma$. The red edges represent a matching. The value of $\mu(q,\sigma)$ is $\frac{q(b) \cdot 3}{\texttt{plu}(b)} = \frac{1}{2}$. In the graph $G^\sigma_{p^\texttt{uni}, q}(a)$, the weight of each edge in the matching is $1/3$. In $\texttt{VDG}(a)$, $w'((2,1)) = w'((3,1)) = 1/12$, $w'((1,2)) = w'((1,3)) = 1/6$, and $w'((3,b)) = w'((2,b)) = 1/4$. The dotted edges in $\texttt{VDG}(a)$ represent the "ill-behaved" portion of the weight.
• Figure 3: A plot of the distortion upper bound guaranteed by $\textsc{LA}_\delta$ as a function of the prediction error $\eta$ for different values of $\delta\in [0,1)$.
• Figure 4: The analogue of Figure \ref{['fig:optimal_trade-off']} for $0$-decisive instances (peer selection setting). On the left, a plot showing the consistency and robustness bounds achieved by $\textsc{LA}_\delta$ as a function of the confidence parameter $\delta$. On the right, a plot capturing the consistency-robustness pairs that can be achieved by $\textsc{LA}_\delta$ for any chosen $\delta \in [0,1)$.

Theorems & Definitions (27)

• Definition 2.1: $(p,q)$-Domination Graph GHS20
• Definition 2.2: Fractional perfect Matching
• Definition 2.3: $(p,q)$-Veto Core KK23
• Definition 3.1: $(p,q)$-algorithms
• Lemma 3.2: munagala2019improvedGHS20
• Lemma 3.3: Ranking-Matching Lemma GHS20
• Theorem 3.4
• Definition 3.5
• Lemma 3.6
• Corollary 3.7