# The Complexity of Constraint Satisfaction Problems * ``` release_nw37mcxkb5g2xmxszh4q3fwfxm ```

by Manuel Bodirsky

Released as a article-journal by Dagstuhl Publishing.

### Abstract

The tractability conjecture for constraint satisfaction problems (CSPs) describes the constraint languages over a finite domain whose CSP can be solved in polynomial-time. The precise formulation of the conjecture uses basic notions from universal algebra. In this talk, we give a short introduction to the universal-algebraic approach to the study of the complexity of CSPs. Finally, we discuss attempts to generalise the tractability conjecture to large classes of constraint languages over infinite domains, in particular for constraint languages that arise in qualitative temporal and spatial reasoning. 1 The Constraint Satisfaction Problem Constraint satisfaction problems are computational problems that can be formalised in several equivalent ways. A mathematically convenient way is to view CSPs as structural homomorphism problems, as follows. Fix a structure Γ with a finite relational signature τ. The domain of Γ need not be finite for the following computational problem to be well-defined. Definition 1 (CSP(Γ)). The constraint satisfaction problem for Γ, denoted by CSP(Γ), is the computational problem to decide for a given finite τ-structure A whether there exists a homomorphism to Γ. The fixed structure Γ is often referred to as the constraint language of the constraint satisfaction problem, since we choose from the relations in Γ to formulate our constraints in the input structure A. We give some concrete examples of CSPs. 1. Graph n-colorability can be formulated as CSP(K n) where K n is the complete loopless graph on n vertices. 2. The question whether a given finite digraph is acyclic, i.e., does not contain a directed cycle, can be formulated as CSP(Q; <). 3. The question whether a given directed graph has a vertex bipartition such that both parts are acyclic can be formulated as CSP(N; E) where E := {(a, b) ∈ N 2 | a < b or (a − b) is odd} .
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