Constraint Satisfaction with Counting Quantifiers

Abstract

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: $\exists^{\geq 1}:=\exists$ and $\exists^{\geq j}$ ($j \neq 1$). Such a CSP is already NP-hard on nonbipartite graph templates. We explore the situation of this generalized CSP on graph templates, giving various conditions for both tractability and hardness. For quantifiers $\exists^{\geq 1}$ and $\exists^{\geq 2}$, we give a dichotomy for all graphs, namely, the problem is NP-hard if the graph contains a triangle or has girth at least 5, and is in P otherwise. We strengthen this result in the following two ways. For bipartite graphs, the problem is in P for forests and graphs of girth 4, and is Pspace-hard otherwise. For complete multipartite graphs, the problem is in L, NP-complete, or Pspace-complete. Finally, using counting quantifiers we solve the complexity of a concrete QCSP whose complexity was previously open.