We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational numbers giving the approximate fraction of constraints that can be satisfied. We prove two new FPC-inexpressibility results for Unique Games: the existence of a $(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within any constant factor. Previous recent work has established similar FPC-inapproximability results for a small handful of other problems. Our construction builds upon some of these ideas, but contains a novel technique. While most FPC-inexpressibility results are based on variants of the CFI-construction, ours is significantly different. We start with a graph of very large girth and label the edges with random affine vector spaces over $\mathbb{F}_2$ that determine the constraints in the two structures. Duplicator's strategy involves maintaining a partial isomorphism over a minimal tree that spans the pebbled vertices of the graph.

翻译：我们研究了在带计数的定点逻辑（FPC）中近似唯一游戏实例最优值的可行性。具体而言，我们证明了FPC解释的准确性下界——这些解释将唯一游戏实例（编码为关系结构）映射为有理数，该有理数表示可满足约束的近似比例。我们为唯一游戏证明了两个新的FPC不可表达性结果：存在一个$(1/2, 1/3 + \delta)$-不可近似间隙，以及任何常数因子内的不可近似性。近期工作已为其他少数问题建立了类似的FPC不可近似性结果。我们的构造借鉴了其中一些思想，但包含了一项新技术。大多数FPC不可表达性结果基于CFI构造的变体，而我们的方法与之显著不同。我们从一张大围长的图出发，用$\mathbb{F}_2$上的随机仿射向量空间标记边，这些向量空间决定了两个结构中的约束。复制者的策略涉及在跨越图中被激活顶点的最小树上维持一个部分同构。