A Rank Lower Bound for Cutting Planes Proofs of Ramsey’s Theorem

@article{Lauria2013ARL,
  title={A Rank Lower Bound for Cutting Planes Proofs of Ramsey’s Theorem},
  author={Massimo Lauria},
  journal={ACM Transactions on Computation Theory (TOCT)},
  year={2013},
  volume={8},
  pages={1 - 13}
}
  • Massimo Lauria
  • Published 8 July 2013
  • Mathematics
  • ACM Transactions on Computation Theory (TOCT)
Ramsey’s Theorem is a cornerstone of combinatorics and logic. In its simplest formulation it says that for every k > 0 and s > 0, there is a minimum number r(k, s) such that any simple graph with at least r(k, s) vertices contains either a clique of size k or an independent set of size s. We study the complexity of proving upper bounds for the number r(k, k). In particular, we focus on the propositional proof system cutting planes; we show that any cutting plane proof of the upper bound “r(k, k… 

References

SHOWING 1-10 OF 31 REFERENCES

A Note on Ramsey Numbers

Rank bounds and integrality gaps for cutting planes procedures

TLDR
This work proves near-optimal rank bounds for Cutting Planes and Lovasz-Schrijver proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas.

Upper and lower bounds for tree-like cutting planes proofs

TLDR
The main result shows that a family of tautologies, introduced in this paper requires exponential-sized tree-like CP proofs, and it follows that tree- like CP cannot polynomially simulate Frege systems.

Asymptotic lower bounds for Ramsey functions

  • J. Spencer
  • Mathematics, Computer Science
    Discret. Math.
  • 1977

Ramsey's Theorem in Bounded Arithmetic

TLDR
It is shown that the finite Ramsey theorem as a Δ0 schema is provable in IΔ0+Ω1 and that propositional formulas expressing the infinite Ramsey theorem have polynomial-size bounded-depth Frege proofs.

The Cutting Plane Proof System with Bounded Degree of Falsity

  • A. Goerdt
  • Mathematics, Computer Science
    CSL
  • 1991
TLDR
This cutting plane proof system for proving the unsatisfiability of propositional formulas in conjunctive normalform is defined, and is the only known system with provably superpolynomial proof size, but polynomial size proofs for the pigeonhole formulas.

Simulating Cutting Plane Proofs with Restricted Degree of Falsity by Resolution

TLDR
It is shown that if the degree of falsity of a Cutting Planes proof Π is bounded by d(n) ≤ n/2, this proof can be easily transformed into a resolution proof of length at most |∏| · (d( n)n−1)64d(n).

On the complexity of cutting-plane proofs