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Many hard examples in exact phase transitions. (English) Zbl 1088.68163

Summary: This paper analyzes the resolution complexity of two random Constraint Satisfaction Problem (CSP) models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it is proved that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CSPs and CNF formulas hard to solve, which can be useful in the experimental evaluation of CSP and SAT algorithms, but also propose models with both many hard instances and exact phase transitions. Finally, conclusions are presented, as well as a detailed comparison of Model RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively, exhibit three different kinds of phase transition behavior in NP-complete problems.

MSC:

68T20 Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.)
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
68Q25 Analysis of algorithms and problem complexity
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