Florent R. Madelaine

Dichotomies for classes of homomorphism problems involving unary functions

We study non-uniform constraint satisfaction problems where the underlying signature contains constant and function symbols as well as relation symbols. Amongst our results are the following. We establish a dichotomy result for the class of non-uniform constraint satisfaction problems over the signature consisting of one unary function symbol by showing that every such problem is either complete for L, via very restricted logical reductions, or trivial (depending upon whether the template function has a fixed point or not). We show that the class of non-uniform constraint satisfaction problems whose templates are structures over the signature λ2 consisting of two unary function symbols reflects the full computational significance of the class of non-uniform constraint satisfaction problems over relational structures. We prove a dichotomy result for the class of non-uniform constraint satisfaction problems where the template is a λ2-structure with the property that the two unary functions involved are the reverse of one another, in that every such problem is either solvable in polynomial-time or NP-complete. Finally, we extend some of our results to the situation where instances of non-uniform constraint satisfaction problems come equipped with lists of elements of the template structure which restrict the set of allowable homomorphisms.

Towards a trichotomy for quantified H -coloring

Hell and Nesetřil proved that the H-colouring problem is NP-complete if, and only if, H is bipartite. In this paper, we investigate the complexity of the quantified H-colouring problem (a restriction of the quantified constraint satisfaction problem to undirected graphs). We introduce this problem using a new two player colouring game. We prove that the quantified H-colouring problem is: 1. tractable, if H is bipartite; 2. NP-complete, if H is not bipartite and not connected; and, 3. Pspace-complete, if H is connected and has a unique cycle, which is of odd length. We conjecture that the last case extends to all non-bipartite connected graphs.

Constraint Satisfaction, Logic and Forbidden Patterns

In the 1990s, Feder and Vardi attempted to find a large subclass of NP which exhibits a dichotomy, that is, where every problem in the subclass is either solvable in polynomial-time or NP-complete. Their studies resulted in a candidate class of problems, namely, those definable in the logic MMSNP. While it remains open as to whether MMSNP exhibits a dichotomy, for various reasons it remains a strong candidate. Feder and Vardi added to the significance of MMSNP by proving that, although MMSNP strictly contains CSP, the class of constraint satisfaction problems, MMSNP and CSP are computationally equivalent. We introduce here a new class of combinatorial problems, the class of forbidden patterns problems FPP, and characterize MMSNP as the finite unions of problems from FPP. We use our characterization to detail exactly those problems that are in MMSNP but not in CSP. Furthermore, given a problem in MMSNP, we are able to decide whether the problem is in CSP or not (this whole process is effective). If the problem is in CSP, then we can construct a template for this problem; otherwise, for any given candidate for the role of template, we can build a counterexample (again, this process is effective).

A Tetrachotomy for Positive First-Order Logic without Equality

We classify completely the complexity of evaluating positive equality-free sentences of first-order logic over a fixed, finite structure D. This problem may be seen as a natural generalisation of the quantified constraint satisfaction problem QCSP(D). We obtain a tetrachotomy for arbitrary finite structures: each problem is either in L, is NP-complete, is co-NP-complete or is P space-complete. Moreover, its complexity is characterised algebraically in terms of the presence or absence of specific surjective hyper-endomorphisms, and, logically, in terms of relativisation properties with respect to positive equality-free sentences. We prove that the meta-problem, to establish for a specific D into which of the four classes the related problem lies, is NP-hard.

The Complexity of Positive First-order Logic without Equality

We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B. This may be seen as a natural generalisation of the non-uniform quantified constraint satisfaction problem QCSP(B). We introduce surjective hyper-endomorphisms and use them in proving a Galois connection that characterises definability in positive equality-free FO. Through an algebraic method, we derive a complete complexity classification for our problems as B ranges over structures of size at most three. Specifically, each problem is either in Logspace, is NP-complete, is coNP-complete or is Pspace-complete.

The Complexity of Positive First-Order Logic without Equality

We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B. This may be seen as a natural generalisation of the non-uniform quantified constraint satisfaction problem QCSP(B). We introduce surjective hyper-endomorphisms and use them in proving a Galois connection that characterises definability in positive equality-free FO. Through an algebraic method, we derive a complete complexity classification for our problems as B ranges over structures of size at most three. Specifically, each problem is either in Logspace, is NP-complete, is coNP-complete or is Pspace-complete.

Universal Structures and the logic of Forbidden Patterns

Forbidden Patterns Problems (FPPs) are a proper generalisation of Constraint Satisfaction Problems (CSPs). However, we show that when the input is connected and belongs to a class which has low tree-depth decomposition (e.g. structure of bounded degree, proper minor closed class and more generally class of bounded expansion) any FPP becomes a CSP. This result can also be rephrased in terms of expressiveness of the logic MMSNP, introduced by Feder and Vardi in relation with CSPs. Our proof generalises that of a recent paper by Nesetril and Ossona de Mendez. Note that our result holds in the general setting of problems over arbitrary relational structures (not just for graphs).

On the complexity of the model checking problem.

The model checking problem for various fragments of first-order logic has attracted much attention over the last two decades: in particular, for the primitive positive and the positive Horn fragments, which are better known as the constraint satisfaction problem and the quantified constraint satisfaction problem, respectively. These two fragments are in fact the only ones for which there is currently no known complexity classification. All other syntactic fragments can be easily classified, either directly or using Schaefers dichotomy theorems for SAT and QSAT, with the exception of the positive equality free fragment. This outstanding fragment can also be classified and enjoys a tetrachotomy: according to the model, the corresponding model checking problem is either tractable, NP-complete, co-NP-complete or Pspace-complete. Moreover, the complexity drop is always witnessed by a generic solving algorithm which uses quantifier relativisation. Furthermore, its complexity is characterised by algebraic means: the presence or absence of specific surjective hyper-operations among those that preserve the model characterise the complexity.

QCSP on Partially Reflexive Cycles - The Wavy Line of Tractability

We study the (non-uniform) quantified constraint satisfaction problem QCSP\((\mathcal{H})\) as \(\mathcal{H}\) ranges over partially reflexive cycles. We obtain a complexity-theoretic dichotomy: QCSP\((\mathcal{H})\) is either in NL or is NP-hard. The separating conditions are somewhat esoteric hence the epithet “wavy line of tractability” (see Figure [5] at end).

Universal structures and the logic of forbidden patterns

Forbidden Patterns Problems (FPPs) are a proper generalisation of Constraint Satisfaction Problems (CSPs). However, we show that when the input belongs to a proper minor closed class, a FPP becomes a CSP. This result can also be rephrased in terms of expressiveness of the logic MMSNP, introduced by Feder and Vardi in relation with CSPs. Our proof generalises that of a recent paper by Nesetřil and Ossona de Mendez. Note that our result holds in the general setting of problems over arbitrary relational structures (not just for graphs).