  We study the problem of finding personalized reserve prices for unit-demand
buyers in multi-unit eager VCG auctions with correlated buyers. The input to
this problem is a dataset of submitted bids of $n$ buyers in a set of auctions.
The goal is to find a vector of reserve prices, one for each buyer, that
maximizes the total revenue across all auctions.
  Roughgarden and Wang (2016) showed that this problem is APX-hard but admits a
greedy $\frac{1}{2}$-approximation algorithm. Later, Derakhshan, Golrezai, and
Paes Leme (2019) gave an LP-based algorithm achieving a $0.68$-approximation
for the (important) special case of the problem with a single-item, thereby
beating greedy. We show in this paper that the algorithm of Derakhshan et al.
in fact does not beat greedy for the general multi-item problem. This raises
the question of whether or not the general problem admits a
better-than-$\frac{1}{2}$ approximation.
  In this paper, we answer this question in the affirmative and provide a
polynomial-time algorithm with a significantly better approximation-factor of
$0.63$. Our solution is based on a novel linear programming formulation, for
which we propose two different rounding schemes. We prove that the best of
these two and the no-reserve case (all-zero vector) is a $0.63$-approximation.
