Valued constraint satisfaction problems (VCSPs) are a large class of computational optimisation problems. If the variables of a VCSP take values from a finite domain, then recent results in constraint satisfaction imply that the problem is in P or NP-complete, depending on the set of admitted cost functions. Here we study the larger class of cost functions over countably infinite domains that have an oligomorphic automorphism group. We present a hardness condition based on a generalisation of pp-constructability as known for (classical) CSPs. We also provide a universal-algebraic polynomial-time tractability condition, based on the concept of fractional polymorphisms. We apply our general theory to study the computational complexity of resilience problems in database theory (under bag semantics). We show how to construct, for every fixed conjunctive query (and more generally for every union of conjunctive queries), a set of cost functions with an oligomorphic automorphism group such that the resulting VCSP is polynomial-time equivalent to the resilience problem; we only require that the query is connected and show that this assumption can be made without loss of generality. For the case where the query is acylic, we obtain a complexity dichotomy of the resilience problem, based on the dichotomy for finite-domain VCSPs. To illustrate the utility of our methods, we exemplarily settle the complexity of a (non-acyclic) conjunctive query whose computational complexity remained open in the literature by verifying that it satisfies our tractability condition. We conjecture that for resilience problems, our hardness and tractability conditions match, which would establish a complexity dichotomy for resilience problems for (unions of) conjunctive queries.

翻译：赋值的约束满足问题（VCSPs）是一大类计算优化问题。当VCSP的变量取有限域值时，约束满足领域的最新结果表明，该问题属于P类或NP完全类，具体取决于所允许的成本函数集合。本文研究具有寡同构自同构群的可数无限域上更大类别的成本函数。我们基于（经典）CSP中已知的pp可构造性推广，提出了一种困难性条件。同时，我们基于分数多态性概念，提供了通用代数多项式时间可解性条件。我们将这一通用理论应用于数据库理论中韧性问题（基于包语义）的计算复杂性研究。我们证明，对于每个固定的合取查询（更一般地，对于每个合取查询的并集），可以构造一个具有寡同构自同构群的成本函数集合，使得所得VCSP与韧性问题在多项式时间内等价；我们仅要求查询是连通的，并证明该假设不失一般性。对于无环查询的情况，基于有限域VCSP的二分性，我们获得了韧性问题的复杂性二分性。为展示方法的实用性，我们通过验证某（非无环）合取查询满足可解性条件，解决了文献中该查询的计算复杂性公开问题。我们猜想：对于韧性问题，困难性条件与可解性条件完全匹配，这将为（合取查询及其并集的）韧性问题建立完整的复杂性二分性。