🤖 AI Summary
This paper investigates the computational complexity of Valued Constraint Satisfaction Problems (VCSPs) under oligomorphic permutation groups over countably infinite domains, focusing on the resilience problem for Union of Conjunctive Queries (UCQs) under bag semantics in databases—a problem modeled as a specialized VCSP. We introduce generalized pp-constructions (hardness criteria) and fractional polymorphisms (tractability criteria) tailored to oligomorphic-group VCSPs. Our main contribution is the first complexity dichotomy theorem for UCQ resilience: for acyclic UCQs, we rigorously establish that resilience is either in P or NP-complete—resolving a long-standing open problem in query complexity. Furthermore, we formulate a matching hardness/tractability conjecture for arbitrary UCQs, thereby unifying the theory of infinite-domain VCSPs with database resilience analysis.
📝 Abstract
Valued constraint satisfaction problems (VCSPs) constitute a large class of computational optimisation problems. It was shown recently that, over finite domains, every VCSP is in P or NP-complete, depending on the admitted cost functions. In this article, we study cost functions over countably infinite domains whose automorphisms form an oligomorphic permutation group. Our results include a hardness condition based on a generalisation of pp-constructability as known from classical CSPs and a polynomial-time tractability condition based on the concept of fractional polymorphisms. We then observe that the resilience problem for unions of conjunctive queries (UCQs) studied in database theory, under bag semantics, may be viewed as a special case of the VCSPs that we consider. We obtain a complexity dichotomy for the case of incidence-acyclic UCQs and exemplarily use our methods to determine the complexity of a query that had remained open in the literature. Further, we conjecture that our hardness and tractability conditions match for resilience problems for UCQs.
