Constraint Satisfaction with Counting Quantifiers 2

Abstract

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 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, in contrast with situations in which the intermediate NP-completeness may appear.Precisely the cases {j}-CSP(K 2j ) are left open here. Of course, {1}-CSP(K 2 ) is graph 2-colorability and is in L, but for j > 1 the situation was very unclear, and the referees noted specifically this lacuna.In this paper we settle this question, and find the surprising situation that {2}-The algorithm for the case {2}-CSP(K 4 ) is specialized and non-trivial, and consists in iteratively constructing a collection of forcing triples where we proceed to look for a contradiction.As a second focus of the paper, we continue the study of {1, 2}-CSP(H). In particular, we focus on finite undirected graphs for which a dichotomy was proposed in [11]. As a fundamental step towards this, we first investigate the complexity of {1, 2}-CSP(P ∞ ), where P ∞ denotes the infinite undirected path. We find tractability here in describing a particular unique obstruction, which takes the form of a special walk, whose presence or absence yields the answer to the problem. Again the algorithm is specialized and non-trivial, and in carefully augmenting it, we construct another polynomial-time algorithm, this time for all finite paths.Theorem 2. {1, 2}-CSP(P n ) is in P, for each n ∈ N.A corollary of this is the following key result. Corollary 3. {1, 2}-CSP(H) is in P, for each forest H.Combined with the results from [9,11], this allows us to observe a dichotomy for {1, 2}-CSP(H) as H ranges over undirected graphs, each problem being either in P or NP-hard, in turn settling a conjecture proposed in [11].In [11], the main preoccupation was in the distinction be...