  We study auction design within the widely acclaimed model of interdependent
values, introduced by Milgrom and Weber [1982]. In this model, every bidder $i$
has a private signal $s_i$ for the item for sale, and a public valuation
function $v_i(s_1,\ldots,s_n)$ which maps every vector of private signals (of
all bidders) into a real value. A recent line of work established the existence
of approximately-optimal mechanisms within this framework, even in the more
challenging scenario where each bidder's valuation function $v_i$ is also
private. This body of work has primarily focused on single-item auctions with
two natural classes of valuations: those exhibiting submodularity over signals
(SOS) and $d$-critical valuations.
  In this work we advance the state of the art on interdependent values with
private valuation functions, with respect to both SOS and $d$-critical
valuations. For SOS valuations, we devise a new mechanism that gives an
improved approximation bound of $5$ for single-item auctions. This mechanism
employs a novel variant of an "eating mechanism", leveraging LP-duality to
achieve feasibility with reduced welfare loss. For $d$-critical valuations, we
broaden the scope of existing results beyond single-item auctions, introducing
a mechanism that gives a $(d+1)$-approximation for any environment with matroid
feasibility constraints on the set of agents that can be simultaneously served.
Notably, this approximation bound is tight, even with respect to single-item
auctions.
