A Gap Trichotomy for Boolean Constraint Problems: Extending Schaefer's Theorem

Author Lucy Ham

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Lucy Ham

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Lucy Ham. A Gap Trichotomy for Boolean Constraint Problems: Extending Schaefer's Theorem. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 36:1-36:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


In this paper, we investigate "gap problems", which are promise problems where YES instances are flexibly satisfiable in a certain sense, and NO instances are not satisfiable at all. These gap problems generalise a family of constraint-related decision problems, including the constraint satisfaction problem itself, the separation problem (can distinct variables be validly assigned distinct values?) and the 2-robust satisfiability problem (does any assignment on two variables extend to a full satisfying assignment?). We establish a Gap Trichotomy Theorem, which on Boolean domains, completely classifies the complexity of the gap problems considered. As a consequence, we obtain several well-known dichotomy results, as well as dichotomies for the separation problem and the 2-robust satisfiability problem: all are either polynomial-time tractable or NP-complete. Schaefer’s original dichotomy is a notable particular case.
  • Constraint Satisfaction Problem
  • Robust satisfiability
  • Clone theory
  • Dichotomy
  • Trichotomy
  • Boolean


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