Constraint Satisfaction with Counting Quantifiers

We initiate the study of

constraint satisfaction problems

(CSPs) in the presence of counting quantifiers, which may be seen as variants of CSPs in the mould of

quantified CSPs

(QCSPs).

We show that a

single

counting quantifier strictly between ∃ 

 ≥ 1

: = ∃ and ∃ 

 ≥ 

n

: = ∀ (the domain being of size

n

) already affords the maximal possible complexity of QCSPs (which have

both

∃ and ∀), being Pspace-complete for a suitably chosen template.

Next, we focus on the complexity of subsets of counting quantifiers on clique and cycle templates. For cycles we give a full trichotomy – all such problems are in L, NP-complete or Pspace-complete. For cliques we come close to a similar trichotomy, but one class remains outstanding.

Afterwards, we consider the generalisation of CSPs in which we augment the extant quantifier ∃ 

 ≥ 1

: = ∃ with the quantifier ∃ 

 ≥ 

j

(

j

 ≠ 1). Such a CSP is already NP-hard on non-bipartite graph templates. We explore the situation of this generalised CSP on bipartite templates, giving various conditions for both tractability and hardness – culminating in a classification theorem for general graphs.

Finally, we use counting quantifiers to solve the complexity of a concrete QCSP whose complexity was previously open.