Manuel Bodirsky

The Complexity of Equality Constraint Languages

Abstract We classify the computational complexity of all constraint satisfaction problems where the constraint language is preserved by all permutations of the domain. A constraint language is preserved by all permutations of the domain if and only if all the relations in the language can be defined by boolean combinations of the equality relation. We call the corresponding constraint languages equality constraint languages. For the classification result we apply the universal-algebraic approach to infinite-valued constraint satisfaction, and show that an equality constraint language is tractable if it admits a constant unary polymorphism or an injective binary polymorphism, and is NP-complete otherwise. We also discuss how to determine algorithmically whether a given constraint language is tractable.

Essential convexity and complexity of semi-algebraic constraints

Let \Gamma be a structure with a finite relational signature and a first-order definition in (R;*,+) with parameters from R, that is, a relational structure over the real numbers where all relation ...

Cores of Countably Categorical Structures

A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for computational complexity classification of constraint satisfaction problems. It is a fundamental fact that every finite structure has a core, i.e., has an endomorphism such that the structure induced by its image is a core; moreover, the core is unique up to isomorphism. Weprove that every \omega -categorical structure has a core. Moreover, every \omega-categorical structure is homomorphically equivalent to a model-complete core, which is unique up to isomorphism, and which is finite or \omega -categorical. We discuss consequences for constraint satisfaction with \omega -categorical templates.

Non-dichotomies in Constraint Satisfaction Complexity

We show that every computational decision problem is polynomial-time equivalent to a constraint satisfaction problem (CSP) with an infinite template. We also construct for every decision problem Lan i¾?-categoricaltemplate Γsuch that Lreduces to CSP(Γ) and CSP(Γ) is in coNPL(i.e., the class coNP with an oracle for L). CSPs with i¾?-categorical templates are of special interest, because the universal-algebraic approach can be applied to study their computational complexity. Furthermore, we prove that there are i¾?-categorical templates with coNP-complete CSPs and i¾?-categorical templates with coNP- intermediate CSPs, i.e., problems in coNP that are neither coNP- complete nor in P (unless P=coNP). To construct the coNP-intermediate CSP with i¾?-categorical template we modify the proof of Ladners theorem. A similar modification allows us to also prove a non-dichotomy result for a class of left-hand side restricted CSPs, which was left open in [10]. We finally show that if the so-called local-global conjecturefor infinite constraint languages(over a finite domain) is false, then there is no dichotomy for the constraint satisfaction problem for infinite constraint languages.

Constraint Satisfaction Problems with Infinite Templates

Allowing templates with infinite domains greatly expands therange of problems that can be formulated as a non-uniformconstraint satisfaction problem. It turns out that many CSPs overinfinite templates can be formulated with templates that areω-categorical. We survey examples of such problems intemporal and spatial reasoning, infinite-dimensional algebra,acyclic colorings in graph theory, artificial intelligence,phylogenetic reconstruction in computational biology, and treedescriptions in computational linguistics. We then give an introduction to the universal-algebraic approachto infinite-domain constraint satisfaction, and discuss how cores,polymorphism clones, and pseudo-varieties can be used to study thecomputational complexity of CSPs with ω-categoricaltemplates. The theoretical results will be illustrated by examplesfrom the mentioned application areas. We close with a series ofopen problems and promising directions of future research.

Minimal functions on the random graph

We show that there is a system of 14 non-trivial finitary functions on the random graph with the following properties: Any non-trivial function on the random graph generates one of the functions of this system by means of composition with automorphisms and by topological closure, and the system is minimal in the sense that no subset of the system has the same property. The theorem is obtained by proving a Ramsey-type theorem for colorings of tuples in finite powers of the random graph, and by applying this to find regular patterns in the behavior of any function on the random graph. As model-theoretic corollaries of our methods we rederive a theorem of Simon Thomas classifying the first-order closed reducts of the random graph, and prove some refinements of this theorem; also, we obtain a classification of the maximal reducts closed under primitive positive definitions, and prove that all reducts of the random graph are model-complete.

Qualitative Temporal and Spatial Reasoning Revisited1

Establishing local consistency is one of the main algorithmic techniques in temporal and spatial reasoning. Acentral question for the various proposed temporal and spatial constraint languages is whether local consistency implies global consistency. Showing that a constraint language Γ has this ‘local-to-global’ property implies polynomial-time tractability of the constraint language, and has further pleasant algorithmic consequences. In the present article, we study the ‘local-to-global’ property by making use of a recently established connection of this property with universal algebra. Roughly speaking, the connection shows that this property is equivalent to the presence of a so-called quasi near-unanimity (QNU) polymorphism of the constraint language. We obtain new algorithmic results and give very concise proofs of previously known theorems. Our results concern well-known and heavily studied formalisms such as the point algebra, Allens interval algebra and the spatial reasoning language RCC-5.

Boltzmann Samplers, Pólya Theory, and Cycle Pointing

We introduce a general method to count unlabeled combinatorial structures and to efficiently generate them at random. The approach is based on pointing unlabeled structures in an “unbiased” way so that a structure of size

Generating labeled planar graphs uniformly at random

n

Constraint Satisfaction with Countable Homogeneous Templates

gives rise to