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

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... e 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} .