Recent years have seen an explosion of interest in lift and project methods, such as those proposed by Lovász and Schrijver [40], Sherali and Adams [49], Balas, Ceria and Cornuejols [6], 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 − ɛ 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. Specifically, after t 2 rounds of Lasserre, the integrality gap decreases to t/(t − 1). Thus, we provide a second example of an integrality gap separation between Lasserre and Sherali Adams. The only other such gap we are aware of is in the recent work of Fernandez de la Vega and Mathieu [19] (respectively of Charikar, Makarychev and Makarychev [12]) showing that the integrality gap for MAXCUT remains 2 − ɛ even after ω(1) (respectively n γ) rounds of Sherali-Adams. On the other hand, it is known that 2 rounds of Lasserre yields a relaxation as least as strong as the Goemans-Williamson SDP, which has an integrality gap of 0.878.

We show that the integrality gap of the standard SDP for VERTEX COVER on instances of n vertices remains 2 − o(1) even after the addition of all hypermetric inequalities. Our lower bound requires new insights into the structure of SDP solutions behaving like ℓ1 metric spaces when one point is removed. We also show that the addition of all ℓ1 inequalities eliminates any solutions that are not convex combination of integral solutions. Consequently, we provide the strongest possible separation between hypermetrics and ℓ1 inequalities with respect to the tightening of the standard SDP for VERTEX COVER.

Abstract. Motivated by applications in combinatorial optimization, we study the extent to which the global properties of a metric space, and especially its embeddability into ℓ1 with low distortion, are determined by the properties of its small subspaces. We establish both upper and lower bounds on the distortion of embedding locally constrained metrics into various target spaces. Other aspects of locally constrained metrics are studied as well, in particular, how far are those metrics from general metrics.