Complexity of Combinations of Qualitative Constraint Satisfaction Problems

Abstract

The CSP of a first-order theory T is the problem of deciding for a given finite set S of atomic formulas whether T ∪ S is satisfiable. Let T 1 and T 2 be two theories with countably infinite models and disjoint signatures. Nelson and Oppen presented conditions that imply decidability (or polynomialtime decidability) of CSP(T 1 ∪ T 2 ) under the assumption that CSP(T 1 ) and CSP(T 2 ) are decidable (or polynomial-time decidable). We show that for a large class of ω-categorical theories T 1 , T 2 the Nelson-Oppen conditions are not only sufficient, but also necessary for polynomial-time tractability of CSP(T 1 ∪ T 2 ) (unless P=NP).

A universal-algebraic proof of the complexity dichotomy for Monotone Monadic SNP

“…We will see that this gives Ramsey order expansions of B ind F , B hom F , and C Φ , too. We need the following general results (from [12]). The structure B is ω-categorical and has no algebraicity.…”

Algebraic Approach to Approximation

“…A further vast enlargement in the combinatorial direction would be provided by incorporating interesting classes of infinite structures. In fact, [21,53] already contribute to this project in the constant factor setting.…”

Tractable Combinations of Theories via Sampling

“…Let T be a theory with finite relational signature. T has the Joint Homomorphism Property (JHP) if for any two models A, B of T there exists a model C of T such that both A and B homomorphically map to C. Proposition 2.3 (Proposition 2.1 in [4]). Let T be a theory with finite relational signature.…”

“…The results of Nelson and Oppen [17,18] provide sufficient conditions for the polynomial-time tractability of CSP(T 1 ∪ T 2 ), covering a great variety of theories. Schulz [19] as well as Bodirsky and Greiner [4] have shown that in many situations, the conditions of Nelson and Oppen are also necessary for polynomial-time tractability (unless P = NP).…”