  Most algorithmic studies on multi-agent information design so far have
focused on the restricted situation with no inter-agent externalities; a few
exceptions investigated truly strategic games such as zero-sum games and
second-price auctions but have all focused only on optimal public signaling.
This paper initiates the algorithmic information design of both \emph{public}
and \emph{private} signaling in a fundamental class of games with negative
externalities, i.e., singleton congestion games, with wide application in
today's digital economy, machine scheduling, routing, etc.
  For both public and private signaling, we show that the optimal information
design can be efficiently computed when the number of resources is a constant.
To our knowledge, this is the first set of efficient \emph{exact} algorithms
for information design in succinctly representable many-player games. Our
results hinge on novel techniques such as developing certain "reduced forms" to
compactly characterize equilibria in public signaling or to represent players'
marginal beliefs in private signaling. When there are many resources, we show
computational intractability results. To overcome the issue of multiple
equilibria, here we introduce a new notion of equilibrium-\emph{oblivious}
hardness, which rules out any possibility of computing a good signaling scheme,
irrespective of the equilibrium selection rule.
