Journal of the ACM (JACM)
Many hard examples for resolution
Journal of the ACM (JACM)
On the complexity of unsatisfiability proofs for random k-CNF formulas
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Short proofs are narrow—resolution made simple
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Setting 2 variables at a time yields a new lower bound for random 3-SAT (extended abstract)
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Space complexity in propositional calculus
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Typical random 3-SAT formulae and the satisfiability threshold
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Bounding the unsatisfiability threshold of random 3-SAT
Random Structures & Algorithms
Simplified and improved resolution lower bounds
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
On the complexity of resolution with bounded conjunctions
Theoretical Computer Science
Narrow proofs may be spacious: separating space and width in resolution
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
A combinatorial characterization of resolution width
Journal of Computer and System Sciences
Towards an optimal separation of space and length in resolution
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
What Is a Real-World SAT Instance?
Proceedings of the 2007 conference on Artificial Intelligence Research and Development
A simplified way of proving trade-off results for resolution
Information Processing Letters
Measuring the hardness of SAT instances
AAAI'08 Proceedings of the 23rd national conference on Artificial intelligence - Volume 1
On minimal unsatisfiability and time-space trade-offs for k-DNF resolution
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
A generating function method for the average-case analysis of DPLL
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
Different approaches to proof systems
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
On the Relative Strength of Pebbling and Resolution
ACM Transactions on Computational Logic (TOCL)
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Relating proof complexity measures and practical hardness of SAT
CP'12 Proceedings of the 18th international conference on Principles and Practice of Constraint Programming
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Some trade-off results for polynomial calculus: extended abstract
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Towards an understanding of polynomial calculus: new separations and lower bounds
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We study the space complexity of refuting unsatisfiable random k-CNFs in the Resolution proof system. We prove that for Δ ≥ 1 and any ε 0, with high probability a random k-CNF over n variables and Δn clauses requires resolution clause space of Ω(n/Δ1 + ε). For constant Δ, this gives us linear, optimal, lower bounds on the clause space. One consequence of this lower bound is the first lower bound for size of treelike resolution refutations of random 3-CNFs with clause density Δ √n. This bound is nearly tight. Specifically, we show that with high probability, a random 3-CNF with Δn clauses requires treelike refutation size of exp(Ω(n/Δ1 + ε). for any ε 0. Our space lower bound is the consequence of three main contributions: (1) We introduce a 2-player Matching Game on bipartite graphs G to prove that there are no perfect matchings in G. (2) We reduce lower bounds for the clause space of a formula F in Resolution to lower bounds for the complexity of the game played on the bipanite graph G(F) associated with F. (3) We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G-PHP, a generalization of the Pigeonhole principle based on bipartite graphs G.