Constraint Satisfaction with Counting Quantifiers 2

  • Barnaby Martin
  • Juraj Stacho
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8476)


We study constraint satisfaction problems (CSPs) in the presence of counting quantifiers ∃  ≥ j , asserting the existence of j distinct witnesses for the variable in question. As a continuation of our previous (CSR 2012) paper [11], we focus on the complexity of undirected graph templates. As our main contribution, we settle the two principal open questions proposed in [11]. Firstly, we complete the classification of clique templates by proving a full trichotomy for all possible combinations of counting quantifiers and clique sizes, placing each case either in P, NP-complete or Pspace-complete. This involves resolution of the cases in which we have the single quantifier ∃  ≥ j on the clique \(\mathbb{K}_{2j}\). Secondly, we confirm a conjecture from [11], which proposes a full dichotomy for ∃ and ∃  ≥ 2 on all finite undirected graphs. The main thrust of this second result is the solution of the complexity for the infinite path which we prove is a polynomial-time solvable problem. By adapting the algorithm for the infinite path we are then able to solve the problem for finite paths, and then trees and forests. Thus as a corollary to this work, combining with the other cases from [11], we obtain a full dichotomy for ∃ and ∃  ≥ 2 quantifiers on finite graphs, each such problem being either in P or NP-hard. Finally, we persevere with the work of [11] in exploring cases in which there is dichotomy between P and Pspace-complete, and contrast this with situations in which the intermediate NP-completeness may appear.


Bipartite Graph Undirected Graph Constraint Satisfaction Total Order Constraint Satisfaction Problem 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Barnaby Martin
    • 1
  • Juraj Stacho
    • 2
  1. 1.School of Science and TechnologyMiddlesex UniversityLondonU.K.
  2. 2.IEOR DepartmentColumbia UniversityNew YorkUnited States

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