Universal algebra and hardness results for constraint satisfaction problems

Benoît Larose, Pascal Tesson
2009 Theoretical Computer Science  
We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem CSP(Γ ) for complexity classes L, NL, P, NP and Mod p L. These criteria also give non-expressibility results for various restrictions of Datalog. Furthermore, we show that if CSP(Γ ) is not first-order definable then it is L-hard. Our proofs rely on tame congruence theory and on a fine-grain analysis of the complexity of reductions used in the algebraic study of CSP. The
more » ... lts pave the way for a refinement of the dichotomy conjecture stating that each CSP(Γ ) lies in P or is NP-complete and they match the recent classification of [E. Allender, M. Bauland, N. Immerman, H. Schnoor, H. Vollmer, The complexity of satisfiability problems: Refining Schaefer's theorem, in: Proc. 30 th Math. Found. of Comp. Sci., MFCS'05, 2005, pp. 71-82] for Boolean CSP. We also infer a partial classification theorem for the complexity of CSP(Γ ) when the associated algebra of Γ is the full idempotent reduct of a preprimal algebra. Constraint satisfaction problems (CSP) provide a unifying framework to study various computational problems arising naturally in artificial intelligence, combinatorial optimization, graph homomorphisms and database theory. An instance of this problem consists of a finite domain, a list of variables and constraints relating the possible values of variables: one has to decide whether the variables can be assigned values that simultaneously satisfy all constraints. This problem is of course NP-complete and so research has focused on identifying tractable subclasses of CSP. A lot of attention has been given to the case where all constraints are constructed from some constraint language Γ , i.e. some set of finitary relations over a fixed domain. In an instance of CSP(Γ ), all constraints are of the form (x i 1 , . . . , x i k ) ∈ R j for some R j ∈ Γ . In their seminal work [15] , Feder and Vardi conjectured that each CSP(Γ ) either lies in P or is NP-complete. This socalled dichotomy conjecture is the natural extension to non-Boolean domains of a celebrated result of Schaefer [28] on the complexity of Generalized Satisfiability which states that CSP(Γ ) is either in P or is NP-complete for any constraint language Γ over the Boolean domain. Progress towards the dichotomy conjecture has been steady over the last fifteen years and has been driven by a number of complementary approaches. One angle of attack relies on universal algebra: there is a natural way to associate to a set of relations Γ an algebra A(Γ ) whose operations are the functions that preserve the relations in Γ and one can show that the complexity of CSP(Γ ) depends on the algebraic structure of A(Γ ). This analysis has led to a number of key results including a verification of the dichotomy conjecture for three-element domains [5] and for so-called list-CSP [6], as well as the identification of wide classes of tractable CSP (see [9] ).
doi:10.1016/j.tcs.2008.12.048 fatcat:qm6drmlez5cy3iqz6xv5cvjazi