Solving the algebraic dichotomy conjecture for constraint satisfaction problems over structures first-order definable in countably infinite finitely bounded homogeneous structures requires understanding the applicability of local-consistency methods in this setting. We study the amount of consistency (measured by relational width) needed to solve CSP for first-order expansions S of countably infinite homogeneous graphs that additionally have bounded strict width, i.e., for which establishing local consistency of an instance of the CSP not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking. Our main result is that the structures S under consideration have relational width exactly (2, L) where L is the maximal size of a forbidden subgraph of a homogeneous graph under consideration, but not smaller than 3. It beats the upper bound (2m, 3m) where m = max(arity(S)+1, L, 3) and arity(S) is the largest arity of a relation in S, which follows from a sufficient condition implying bounded relational width from the literature. Since L may be arbitrarily large, our result contrasts the collapse of the relational bounded width hierarchy for finite structures , whose relational width, if finite, is always at most (2,3).

中文翻译：

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有界严格宽度齐次图的一阶展开的关系宽度

解决代数二分法猜想的约束满足问题在一阶可数无限有限有界齐次结构中定义的结构需要了解局部一致性方法在这种情况下的适用性。我们研究了解决可数无限齐次图的一阶展开 S 所需的一致性量（由关系宽度测量），这些图还具有严格的宽度，即，建立 CSP 实例的局部一致性不仅决定如果有一个解决方案，还要确保每个解决方案都可以通过贪婪地为变量分配值来从本地一致的实例中获得，而无需回溯。我们的主要结果是所考虑的结构 S 的关系宽度恰好为 (2, L) 其中 L 是所考虑的同构图的禁止子图的最大尺寸，但不小于 3。它超过上限 (2m, 3m) 其中 m = max(arity(S)+1, L, 3 ) 和 arity(S) 是 S 中关系的最大数量，这是从文献中暗示有界关系宽度的充分条件得出的。由于 L 可能是任意大的，我们的结果与有限结构的关系有界宽度层次结构的崩溃形成对比，其关系宽度，如果有限，总是至多 (2,3)。