Tree clustering for constraint networks (research note)
Artificial Intelligence
Improvements to propositional satisfiability search algorithms
Improvements to propositional satisfiability search algorithms
Unit Refutations and Horn Sets
Journal of the ACM (JACM)
On programming of arithmetic operations
Communications of the ACM
A comparison of structural CSP decomposition methods
Artificial Intelligence
Information and Computation
Simplified and improved resolution lower bounds
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Space complexity of random formulae in resolution
Random Structures & Algorithms
Towards understanding and harnessing the potential of clause learning
Journal of Artificial Intelligence Research
Backdoors to typical case complexity
IJCAI'03 Proceedings of the 18th international joint conference on Artificial intelligence
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Frequently, in the SAT and CSP communities, people talk about real-world problems, without any formal or precise definition of what “real-world” means. This notion is used as opposed to randomly generated problems and theoretical combinatorial principles, like the pigeonhole. People agree that modern SAT and CSP solvers perform well in these real-world problems with a hidden structure, and more so as more intelligent are the strategies used on them. Here we define a formal notion, called the Strahler number, that measures the degree of structure of an unsatisfiable SAT instance. We argue why this notion corresponds to the informal idea of real-world problem. If a formula has an small Strahler number, then it has a lot of structure, and it is easy to prove it, even if it has many variables. We prove that there is a SAT solver, the Beame-Pitassi algorithm [2], that works on time O(ns), being n the number of variables, and s the Strahler of the formula. We also show that Horn and 2-SAT unsatisfiable formulas, that can be solved in polynomial time, have Strahler number 1 and 2 respectively. We compare the Strahler number with other notions defined with the same purpose like backdoors [15], and prove that the Strahler number can be arbitrarily smaller than the size of strong backdoors. We show the same relation for the size of cycle-cutsets and the treewidth in tree decompositions. Finally, we compare with the space of resolution calculus, defined for a different purpose.