We consider bounded width CNF-formulas where the width is measured by popular graph width measures on graphs associated to CNF-formulas. Such restricted graph classes, in particular those of bounded treewidth, have been extensively studied for their uses in the design of algorithms for various computational problems on CNF-formulas. Here we consider the expressivity of these formulas in the model of clausal encodings with auxiliary variables. We first show that bounding the width for many of the measures from the literature leads to a dramatic loss of expressivity, restricting the formulas to such of low communication complexity. We then show that the width of optimal encodings with respect to different measures is strongly linked: there are two classes of width measures, one containing primal treewidth and the other incidence cliquewidth, such that in each class the width of optimal encodings only differs by constant factors. Moreover, between the two classes the width differs at most by a factor logarithmic in the number of variables. Both these results are in stark contrast to the setting without auxiliary variables where all width measures we consider here differ by more than constant factors and in many cases even by linear factors.

中文翻译：

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带有辅助变量的 CNF 编码的图形宽度度量

我们考虑有界宽度 CNF 公式，其中宽度是通过与 CNF 公式相关的图上的流行图宽度度量来测量的。这种受限制的图类，特别是那些有界树宽的图类，因其在 CNF 公式上的各种计算问题的算法设计中的用途而被广泛研究。在这里，我们考虑这些公式在带有辅助变量的条款编码模型中的表达能力。我们首先表明，限制文献中许多度量的宽度会导致表现力的显着损失，将公式限制为低通信复杂性。然后，我们证明了关于不同度量的最佳编码的宽度是密切相关的：有两类宽度度量，一类包含原始树宽度，另一类包含关联集团宽度，这样在每个类别中，最佳编码的宽度仅因常数因子而异。此外，在两个类别之间，宽度最多相差一个变量数量的对数因子。这两个结果都与没有辅助变量的设置形成鲜明对比，其中我们在这里考虑的所有宽度度量的差异不仅仅是常数因子，在许多情况下甚至是线性因子。