Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Exponential lower bounds and integrality gaps for tree-like Lovász-Schrijver procedures
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Sherali-adams relaxations of the matching polytope
Proceedings of the forty-first annual ACM symposium on Theory of computing
A better approximation ratio for the vertex cover problem
ACM Transactions on Algorithms (TALG)
Optimal Sherali-Adams Gaps from Pairwise Independence
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Integrality gaps of linear and semi-definite programming relaxations for Knapsack
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Integrality Gaps of $2-o(1)$ for Vertex Cover SDPs in the Lovász-Schrijver Hierarchy
SIAM Journal on Computing
Sherali-Adams relaxations and indistinguishability in counting logics
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Exponential Lower Bounds and Integrality Gaps for Tree-Like Lovász-Schrijver Procedures
SIAM Journal on Computing
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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.