In this paper we study the interactions between so-called fractional relaxations of the integer programs (IPs) which encode homomorphism and isomorphism of relational structures. We give a combinatorial characterization of a certain natural linear programming (LP) relaxation of homomorphism in terms of fractional isomorphism. As a result, we show that the families of constraint satisfaction problems (CSPs) that are solvable by such linear program are precisely those that are closed under an equivalence relation which we call Weisfeiler-Leman invariance. We also generalize this result to the much broader framework of Promise Valued Constraint Satisfaction Problems, which brings together two well-studied extensions of the CSP framework. Finally, we consider the hierarchies of increasingly tighter relaxations of the homomorphism and isomorphism IPs obtained by applying the Sherali-Adams and Weisfeiler-Leman methods respectively. We extend our combinatorial characterization of the basic LP to higher levels of the Sherali-Adams hierarchy, and we generalize a well-known logical characterization of the Weisfeiler-Leman test from graphs to relational structures.

翻译：本文研究编码关系结构同态与同构的整数规划（IP）的所谓分数松弛之间的相互作用。我们给出了同态的一种自然线性规划（LP）松弛的分数同构组合刻画。结果表明，此类线性规划可解的约束满足问题（CSP）族恰好是那些在被称为Weisfeiler-Leman不变性的等价关系下封闭的问题族。我们还将这一结果推广到更广泛的承诺值约束满足问题框架，该框架融合了CSP框架的两个公认扩展。最后，我们研究了分别通过Sherali-Adams方法和Weisfeiler-Leman方法获得的同态与同构IP的逐层紧缩松弛层次。我们将基本LP的组合刻画扩展至Sherali-Adams层次的高层，并将Weisfeiler-Leman检验从图到关系结构的经典逻辑刻画进行了推广。