扩展Monadic二阶逻辑在受限图类上的表达能力
Expanding the expressive power of Monadic Second-Order logic on restricted graph classes
摘要：
我们结合整数线性规划和一元二阶模型检验的最新进展，得到两个新的算法元定理的有界顶点覆盖图。首先表明，cardMSO 1，著名的Monadic二阶逻辑的扩展，通过增加基数约束，可以解决FPT时间参数化的顶点覆盖。第二个元定理表明，由Rao引入的MSO划分问题也可以在FPT时间内解决相同的参数。我们的贡献的意义源于这样一个事实，即这些形式主义可以描述的问题，这是W[1]-困难，甚至NP-困难的有界树宽度的图。此外，我们的算法只有一个基本的依赖于参数和公式。我们还表明，这两个结果很容易从顶点覆盖扩展到邻域多样性。
英文摘要：
We combine integer linear programming and recent advances in Monadic Second-Order model checking to obtain two new algorithmic meta-theorems for graphs of bounded vertex-cover. The first shows that cardMSO1, an extension of the well-known Monadic Second-Order logic by the addition of cardinality constraints, can be solved in FPT time parameterized by vertex cover. The second meta-theorem shows that the MSO partitioning problems introduced by Rao can also be solved in FPT time with the same parameter. The significance of our contribution stems from the fact that these formalisms can describe problems which are W[1]-hard and even NP-hard on graphs of bounded tree-width. Additionally, our algorithms have only an elementary dependence on the parameter and formula. We also show that both results are easily extended from vertex cover to neighborhood diversity.
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