Journal of the ACM (JACM)
On the complexity of cutting-plane proofs
Discrete Applied Mathematics
Proceedings of the first Malta conference on Graphs and combinatorics
Lower bounds for cutting planes proofs with small coefficients
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Using the Groebner basis algorithm to find proofs of unsatisfiability
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Proof complexity in algebraic systems and bounded depth Frege systems with modular counting
Computational Complexity
On a Representation of the Matching Polytope Via Semidefinite Liftings
Mathematics of Operations Research
Lower bounds for the polynomial calculus and the Gröbner basis algorithm
Computational Complexity
Lower bounds for the polynomial calculus
Computational Complexity
On the Chvátal rank of polytopes in the 0/1 cube
Discrete Applied Mathematics
Linear gaps between degrees for the polynomial calculus modulo distinct primes
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Linear lower bound on degrees of positivstellensatz calculus proofs for the parity
Theoretical Computer Science
Complexity of Positivstellensatz proofs for the knapsack
Computational Complexity
When Does the Positive Semidefiniteness Constraint Help in Lifting Procedures?
Mathematics of Operations Research
Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube
Proceedings of the 7th International IPCO Conference on Integer Programming and Combinatorial Optimization
On the Matrix-Cut Rank of Polyhedra
Mathematics of Operations Research
Exponential Lower Bound for Static Semi-algebraic Proofs
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Exponential Lower Bounds on the Lengths of Some Classes of Branch-and-Cut Proofs
Mathematics of Operations Research
Tight rank lower bounds for the Sherali–Adams proof system
Theoretical Computer Science
Hardness amplification in proof complexity
Proceedings of the forty-second ACM symposium on Theory of computing
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
Complexity of propositional proofs
CSR'10 Proceedings of the 5th international conference on Computer Science: theory and Applications
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Proof systems for polynomial inequalities in 0-1 variables include the well-studied Cutting Planes proof system (CP) and the Lov谩sz-Schrijver calculi (LS) utilizing linear, respectively, quadratic, inequalities. We introduce generalizations LSd of LS involving polynomial inequalities of degree at most d.Surprisingly, the systems LSd turn out to be very strong. We construct polynomial-size bounded degree LSd proofs of the clique-coloring tautologies (which have no polynomial-size CP proofs), the symmetric knapsack problem (which has no bounded degree Positivstellensatz Calculus (PC) proofs), and Tseitin's tautologies (hard for many known proof systems). Extending our systems with a division rule yields a polynomial simulation of CP with polynomially bounded coefficients, while other extra rules further reduce the proof degrees for the aforementioned examples.Finally, we prove lower bounds on Lov谩sz-Schrijver ranks, demonstrating, in particular, their rather limited applicability for proof complexity.