In 1996, Bodlaender showed the celebrated result that an optimal tree decomposition of a graph of bounded treewidth can be found in linear time. The algorithm is based on an algorithm of Bodlaender and Kloks that computes an optimal tree decomposition given a non-optimal tree decomposition of bounded width. Both algorithms, in particular the second, are hardly accessible. In our review, we present them in a much simpler way than the original presentations. In our description of the second algorithm, we start by explaining how all tree decompositions of subtrees defined by the nodes of the given tree decomposition can be enumerated. We group tree decompositions into equivalence classes depending on the current node of the given tree decomposition, such that it suffices to enumerate one tree decomposition per equivalence class and, for each node of the given tree decomposition, there are only a constant number of classes which can be represented in constant space. Our description of the first algorithm further simplifies Perkovic and Reed's simplification.

中文翻译：

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重新审视最优树分解：一种更简单的线性时间 FPT 算法

1996 年，Bodlaender 展示了著名的结果，即可以在线性时间内找到有界树宽图的最优树分解。该算法基于 Bodlaender 和 Kloks 的算法，该算法在给定有界宽度的非最优树分解的情况下计算最优树分解。这两种算法，尤其是第二种算法，都难以访问。在我们的评论中，我们以比原始演示更简单的方式呈现它们。在我们对第二种算法的描述中，我们首先解释了如何枚举由给定树分解的节点定义的子树的所有树分解。我们根据给定树分解的当前节点将树分解分组为等价类，这样就足以为每个等价类枚举一个树分解，并且，对于给定树分解的每个节点，只有恒定数量的类可以在恒定空间中表示。我们对第一个算法的描述进一步简化了 Perkovic 和 Reed 的简化。