关于一元二阶逻辑的荆棘、格状子元和参数化难解性
On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic
摘要：
荆棘是作为树宽的对偶概念引入的，树宽是罗伯逊和西摩的图子式理论中最核心的概念之一。最近，Grohe和Marx证明了在|G|中存在图G，其中每一个阶大于树宽平方根的荆棘都是指数大小的.在积极的一面，他们表明存在的多项式大小的荆棘的平方根的树宽的顺序，日志因素。我们提供了第一个多项式时间算法来构建一个荆棘一般的图形，并实现这个界限，对数因子。我们用这个算法来构建网格状的未成年人，最近由里德和伍德介绍的网格未成年人的替代结构，在多项式时间。使用网格状的未成年人，我们介绍了一个完美的荆棘的概念和算法，找到一个在多项式时间。完美的荆棘是荆棘与一个特别简单的结构，他们也为我们提供了一个子图，有界的程度和仍然大的树宽，我们使用它们来获得一个元定理，决定某些参数化的子图闭问题的一般图在时间单指数的参数。我们的工作的第二部分涉及提供一个下界Courcelle的著名定理，指出，每一个图形属性，可以表示一个句子在一元二阶逻辑（MSO），可以决定由一个线性时间算法类图的有界树宽。利用我们的结果，从我们的工作的第一部分，我们建立了一个强下界的易处理的MSO类的着色图。
英文摘要：
Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger than the square root of the treewidth is of exponential size in |G|. On the positive side, they show the existence of polynomial-sized brambles of the order of the square root of the treewidth, up to log factors. We provide the first polynomial time algorithm to construct a bramble in general graphs and achieve this bound, up to log-factors. We use this algorithm to construct grid-like minors, a replacement structure for grid-minors recently introduced by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce the notion of a perfect bramble and an algorithm to find one in polynomial time. Perfect brambles are brambles with a particularly simple structure and they also provide us with a subgraph that has bounded degree and still large treewidth; we use them to obtain a meta-theorem on deciding certain parameterized subgraph-closed problems on general graphs in time singly exponential in the parameter. The second part of our work deals with providing a lower bound to Courcelle's famous theorem, stating that every graph property that can be expressed by a sentence in monadic second-order logic (MSO), can be decided by a linear time algorithm on classes of graphs of bounded treewidth. Using our results from the first part of our work we establish a strong lower bound for tractability of MSO on classes of colored graphs.
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