Lovasz and Schrijver [13] defined three progressively stronger procedures LS0, LS and LS+, for systematically tightening linear relaxations over many rounds. All three procedures yield the integral hull after at most n rounds. On the other hand, constant rounds of LS_+ can derive the relaxations behind many famous approximation algorithms such as those for MAX-CUT, SPARSEST-CUT. So proving round lower bounds for these procedures on specific problems may give evidence about inapproximability.

We prove new round lower bounds for VERTEX COVER in the LS hierarchy. Arora et al. [3] showed that the integrality gap for VERTEX COVER relaxations remains 2.o(1) even after \Omega(log n) rounds LS. However, their method can only prove round lower bounds as large as the girth of the input graph, which is O(log n) for interesting graphs.

We break through this "girth barrier" and show that the integrality gap for VERTEX COVER remains 1.5 . \in even after \Omega(log^2 n) rounds of LS. In contrast, the best PCPbased results only rule out 1.36-approximations. Moreover, we conjecture that the new technique we introduce to prove our lower bound, the "fence" method, may lead to linear or nearly linear LS round lower bounds for VERTEX COVER.