We study various SDP formulations for {\sc Vertex Cover} by adding different
constraints to the standard formulation. We show that {\sc Vertex Cover} cannot
be approximated better than 2−o(1) even when we add the so called pentagonal
inequality constraints to the standard SDP formulation, en route answering an
open question of Karakostas~\cite{Karakostas}. We further show the surprising
fact that by strengthening the SDP with the (intractable) requirement that the
metric interpretation of the solution is an ℓ1 metric, we get an exact
relaxation (integrality gap is 1), and on the other hand if the solution is
arbitrarily close to being ℓ1 embeddable, the integrality gap may be as
big as 2−o(1). Finally, inspired by the above findings, we use ideas from the
integrality gap construction of Charikar \cite{Char02} to provide a family of
simple examples for negative type metrics that cannot be embedded into ℓ1
with distortion better than 8/7-\eps. To this end we prove a new
isoperimetric inequality for the hypercube.Comment: A more complete version. Changed order of results. A complete proof
of (current) Theorem