The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that many Max-CSPs remain as hard to approximate as in the general case even when the factor graph is fixed (depending only on the size of the instance) and known in advance.

Examples of results obtained for this restricted setting are:

(1)	Optimal inapproximability for Max-3-Lin and Max-3-Sat (Håstad, J. ACM 2001).
(2)	Approximation resistance for predicates supporting pairwise independent subgroups (Chan, J. ACM 2016).
(3)	Hardness of the “(2 + ϵ)-Sat” problem and other Promise CSPs (Austrin et al., SIAM J. Comput. 2017).

The main technical tool used to establish these results is a new way of folding the long code which we call “functional folding”.

中文翻译：

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通用因子图的最优不近似性

约束满足问题 (CSP) 实例的因子图是二分图，指示每个约束中出现哪些变量。 CSP 的实例由因子图以及应用于每个约束的谓词列表给出。我们发现，即使因子图是固定的（仅取决于实例的大小）并且提前已知，许多 Max-CSP 仍然像一般情况一样难以近似。

此限制设置获得的结果示例如下：

(1)	Max-3-Lin 和 Max-3-Sat 的最佳不逼近性（Håstad, J. ACM 2001）。
(2)	支持成对独立子组的谓词的近似阻力（Chan, J. ACM 2016）。
(3)	“(2 + ϵ)-Sat”问题和其他 Promise CSP 的硬度（Austrin 等人，SIAM J.Comput.2017）。

用于建立这些结果的主要技术工具是一种折叠长代码的新方法，我们称之为“功能折叠”。