For a constraint satisfaction problem (CSP), a robust satisfaction algorithm is one that outputs an assignment satisfying most of the constraints on instances that are near-satisfiable. It is known that the CSPs that admit efficient robust satisfaction algorithms are precisely those of bounded width, i.e., CSPs whose satisfiability can be checked by a simple local consistency algorithm (eg., 2-SAT or Horn-SAT in the Boolean case). While the exact satisfiability of a bounded width CSP can be checked by combinatorial algorithms, the robust algorithm is based on rounding a canonical Semidefinite programming(SDP) relaxation. In this work, we initiate the study of robust satisfaction algorithms for promise CSPs, which are a vast generalization of CSPs that have received much attention recently. The motivation is to extend the theory beyond CSPs, as well as to better understand the power of SDPs. We present robust SDP rounding algorithms under some general conditions, namely the existence of particular high-dimensional Boolean symmetries known as majority or alternating threshold polymorphisms. On the hardness front, we prove that the lack of such polymorphisms makes the PCSP hard for all pairs of symmetric Boolean predicates. Our method involves a novel method to argue SDP gaps via the absence of certain colorings of the sphere, with connections to sphere Ramsey theory. We conjecture that PCSPs with robust satisfaction algorithms are precisely those for which the feasibility of the canonical SDP implies (exact) satisfiability. We also give a precise algebraic condition, known as a minion characterization, of which PCSPs have the latter property.

翻译：对于约束满足问题（CSP），鲁棒满足算法是指在近乎可满足的实例上，输出满足大部分约束的赋值算法。已知允许高效鲁棒满足算法的CSP正是那些有界宽度的CSP，即其可满足性可通过简单的局部一致性算法（例如布尔情况下的2-SAT或Horn-SAT）验证的CSP。尽管有界宽度CSP的精确可满足性可通过组合算法验证，但鲁棒算法基于对规范半定规划（SDP）松弛的舍入。本文中，我们首次研究Promise CSP（一种近期备受关注的CSP的广泛推广）的鲁棒满足算法。其动机既在于将理论拓展至CSP之外，也在于更深入理解SDP的威力。我们在一般条件下提出了鲁棒的SDP舍入算法，这些条件具体指存在特定的高维布尔对称性（多数或交替阈值多态）。在困难性方面，我们证明：对于所有对称布尔谓词对，缺乏此类多态性将导致PCSP困难。我们的方法涉及一种通过球面特定着色的缺失来论证SDP间隙的新颖方法，并与球面拉姆齐理论相关联。我们推测：具有鲁棒满足算法的PCSP正是那些规范SDP的可行性蕴含（精确）可满足性的PCSP。我们还给出了PCSP具备后一性质的精确代数条件（称为小从特征刻画）。