Towards strong nonapproximability results in the Lovasz-Schrijver hierarchy
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Rank complexity gap for Lovász-Schrijver and Sherali-Adams proof systems
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Annals of Mathematics and Artificial Intelligence
Linear programming relaxations of maxcut
SODA '07 Proceedings of the eighteenth 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
Hardness amplification in proof complexity
Proceedings of the forty-second ACM symposium on Theory of computing
Improving integrality gaps via Chvátal-Gomory rounding
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on 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
Lower bounds for lovász-schrijver systems and beyond follow from multiparty communication complexity
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Towards optimal integrality gaps for hypergraph vertex cover in the lovász-schrijver hierarchy
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
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We present a new method for proving rank lower bounds for Cutting Planes (CP) and several procedures based on lifting due to Lovász and Schrijver (LS), when viewed as proof systems for unsatisfiability. We apply this method to obtain the following new results: First, we prove near-optimal rank bounds for Cutting Planes and Lovász-Schrijver proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas. It follows from these lower bounds that a linear number of rounds of CP or LS procedures when applied to relaxations of integer linear programs is not sufficient for reducing the integrality gap. Secondly, we give unsatisfiable examples that have constant rank CP and LS proofs but that require linear rank Resolution proofs. Thirdly, we give examples where the CP rank is O(log n) but the LS rank is linear. Finally, we address the question of size versus rank: we show that, for both proof systems, rank does not accurately reflect proof size. Specifically, there are examples with polynomial-size CP/LS proofs, but requiring linear rank.