Constraint Satisfaction with Counting Quantifiers

TL;DR

This paper explores the complexity of constraint satisfaction problems with counting quantifiers, revealing maximal complexity with a single counting quantifier and providing a comprehensive classification for various graph templates.

Contribution

It introduces the study of CSPs with counting quantifiers, establishing complexity classifications and solving an open problem in QCSP complexity.

Findings

Single counting quantifier can lead to Pspace-complete problems.

Complete trichotomy for cycle templates: L, NP-complete, or Pspace-complete.

Classification theorem for generalized CSPs on graphs.

Abstract

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 exists^1:=exists and exists^n:=forall (the domain being of size n) already affords the maximal possible complexity of QCSPs (which have both exists and forall), 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 case remains outstanding. Afterwards, we consider the generalisation of CSPs in which we augment the extant quantifier exists^1:=exists with the quantifier exists^j (j not 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.