Many hard examples for resolution
Journal of the ACM (JACM)
A sharp threshold for k-colorability
Random Structures & Algorithms
2+p-SAT: relation of typical-case complexity to the nature of the phase transition
Random Structures & Algorithms - Special issue on statistical physics methods in discrete probability, combinatorics, and theoretical computer science
Artificial Intelligence
The phase transition in 1-in-k SAT and NAE 3-SAT
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Short proofs are narrow—resolution made simple
Journal of the ACM (JACM)
Propositional proof complexity: past, present, and future
Current trends in theoretical computer science
Statistical mechanics methods and phase transitions in optimizationproblems
Theoretical Computer Science - Phase transitions in combinatorial problems
Frozen development in graph coloring
Theoretical Computer Science - Phase transitions in combinatorial problems
Models and thresholds for random constraint satisfaction problems
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
The Efficiency of Resolution and Davis--Putnam Procedures
SIAM Journal on Computing
Resolution Complexity of Random Constraints
CP '02 Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming
The Resolution Complexity of Random Constraint Satisfaction Problems
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Combinatorial sharpness criterion and phase transition classification for random CSPs
Information and Computation
Threshold properties of random boolean constraint satisfaction problems
Discrete Applied Mathematics - Special issue: Typical case complexity and phase transitions
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We study the connection between the order of phase transitions in combinatorial problems and the complexity of decision algorithms for such problems. We rigorously show that, for a class of random constraint satisfaction problems, a limited connection between the two phenomena indeed exists. Specifically, we extend the definition of the spine order parameter of Bollobás et al. [10] to random constraint satisfaction problems, rigorously showing that for such problems a discontinuity of the spine is associated with a 2驴(n) resolution complexity (and thus a 2驴(n) complexity of DPLL algorithms) on random instances. The two phenomena have a common underlying cause: the emergence of "large" (linear size) minimally unsatisfiable subformulas of a random formula at the satisfiability phase transition.We present several further results that add weight to the intuition that random constraint satisfaction problems with a sharp threshold and a continuous spine are "qualitatively similar to random 2-SAT". Finally, we argue that it is the spine rather than the backbone parameter whose continuity has implications for the decision complexity of combinatorial problems, and we provide experimental evidence that the two parameters can behave in a different manner.