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：估值约束满足问题（VCSPs）是一大类计算优化问题。若VCSP的变量取值于有限域，则约束满足领域的最新结果表明，根据所允许的成本函数集合，该问题属于P类或NP完全问题。本文研究具有寡同构自同构群的无限可数域上更广义的成本函数类。我们基于经典CSP中已知的pp-可构造性的推广，提出了一个困难性条件。同时，我们利用分数多态性概念，给出了一个泛代数意义下的多项式时间可处理性条件。我们将这一通用理论应用于数据库理论中（基于包语义的）韧性问题的计算复杂性研究。我们证明：对于每个固定合取查询（更一般地，对于每个合取查询的并集），可构造一个具有寡同构自同构群的成本函数集，使得所得的VCSP在多项式时间意义上等价于韧性问题；我们仅要求查询是连通的，并证明该假设不失一般性。当查询为无环查询时，基于有限域VCSP的二分性，我们得到了韧性问题的复杂性二分性。为展示本方法的实用性，我们以文献中计算复杂性尚待解决的（非无环）合取查询为例，验证其满足本文的可处理性条件，从而确定了该查询的复杂性。我们猜想：对于韧性问题，本文的困难性与可处理性条件相匹配，这将为（合取查询及其并集的）韧性问题建立复杂性二分性。