Summary: Integrality Gaps of Linear and Semi-definite Programming
Relaxations for Knapsack
Anna R. Karlin
C. Thach Nguyen
Recent years have seen an explosion of interest in lift and project methods, such as those proposed
by Lov´asz and Schrijver , Sherali and Adams , Balas, Ceria and Cornuejols , Lasserre [36, 37]
and others. These methods are systematic procedures for constructing a sequence of increasingly tight
mathematical programming relaxations for 0-1 optimization problems.
One major line of research in this area has focused on understanding the strengths and limitations
of these procedures. Of particular interest to our community is the question of how the integrality gaps
for interesting combinatorial optimization problems evolve through a series of rounds of one of these
procedures. On the one hand, if the integrality gap of successive relaxations drops sufficiently fast, there
is the potential for an improved approximation algorithm. On the other hand, if the integrality gap for
a problem persists, this can be viewed as a lower bound in a certain restricted model of computation.
In this paper, we study the integrality gap in these hierarchies for the knapsack problem. We have
two main results. First, we show that an integrality gap of 2 - persists up to a linear number of
rounds of Sherali-Adams. This is interesting, since it is well known that knapsack has a fully polynomial
time approximation scheme [30, 39]. Second, we show that Lasserre's hierarchy closes the gap quickly.