Constraint Satisfaction with Counting Quantifiers

We initiate the study of constraint satisfaction problems (CSPs) in the presence of counting quantifiers $\exists^{\geq j}$ which assert the existence of at least $j$ elements such that the ensuing property holds. These are natural variants of CSPs in the mould of quantified CSPs (QCSPs). Namely, $\exists^{\geq 1}:=\exists$ and $\exists^{\geq n}:=\forall$ (for the domain of size $n$). We observe that a single counting quantifier $\exists^{\geq j}$ strictly between $\exists$ and $\forall$ already affords the maximal possible complexity of QCSPs (which have both $\exists$ and $\forall$), namely, being Pspace-complete for a suitably chosen template. Therefore, to better understand the complexity of this problem, we focus on restricted cases for which we derive the following results. First, for all subsets of counting quantifiers on clique and cycle templates, we give a full trichotomy---all such problems are in P, NP-complete, or Pspace-complete. Second, we consider the problem with exactly two quantifiers: ...

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