We study integrality gap (IG) lower bounds on strong LP and SDP relaxations
derived by the Sherali-Adams (SA), Lovasz-Schrijver-SDP (LS+), and
Sherali-Adams-SDP (SA+) lift-and-project (L&P) systems for the
t-Partial-Vertex-Cover (t-PVC) problem, a variation of the classic Vertex-Cover
problem in which only t edges need to be covered. t-PVC admits a
2-approximation using various algorithmic techniques, all relying on a natural
LP relaxation. Starting from this LP relaxation, our main results assert that
for every epsilon > 0, level-Theta(n) LPs or SDPs derived by all known L&P
systems that have been used for positive algorithmic results (but the Lasserre
hierarchy) have IGs at least (1-epsilon)n/t, where n is the number of vertices
of the input graph. Our lower bounds are nearly tight.
Our results show that restricted yet powerful models of computation derived
by many L&P systems fail to witness c-approximate solutions to t-PVC for any
constant c, and for t = O(n). This is one of the very few known examples of an
intractable combinatorial optimization problem for which LP-based algorithms
induce a constant approximation ratio, still lift-and-project LP and SDP
tightenings of the same LP have unbounded IGs.
We also show that the SDP that has given the best algorithm known for t-PVC
has integrality gap n/t on instances that can be solved by the level-1 LP
relaxation derived by the LS system. This constitutes another rare phenomenon
where (even in specific instances) a static LP outperforms an SDP that has been
used for the best approximation guarantee for the problem at hand. Finally, one
of our main contributions is that we make explicit of a new and simple
methodology of constructing solutions to LP relaxations that almost trivially
satisfy constraints derived by all SDP L&P systems known to be useful for
algorithmic positive results (except the La system).Comment: 26 page