  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 $p\in \Delta(V)$ and $q\in
\Delta(C)$ 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.
