The classic algorithm of Bodlaender and Kloks [J. Algorithms, 1996] solves the following problem in linear fixed-parameter time: given a tree decomposition of a graph of (possibly suboptimal) width $k$, compute an optimum-width tree decomposition of the graph. In this work, we prove that this problem can also be solved in MSO in the following sense: for every positive integer $k$, there is an MSO transduction from tree decompositions of width $k$ to tree decompositions of optimum width. Together with our recent results [LICS 2016], this implies that for every $k$ there exists an MSO transduction which inputs a graph of treewidth $k$, and nondeterministically outputs its tree decomposition of optimum width. We also show that MSO transductions can be implemented in linear fixed-parameter time, which enables us to derive the algorithmic result of Bodlaender and Kloks as a corollary of our main result.

翻译：Bodlaender和Kloks的经典算法[J. Algorithms, 1996] 解决了线性固定参数时间的下列问题:考虑到树上对(可能低于最佳)宽度$k$的图形进行分解,计算出该图的最佳宽度树分解。在这项工作中,我们证明这个问题也可以在MOO中从以下意义上加以解决:对于每个正整价美元来说,从宽度$k的树分解到最佳宽度的树分解,有一个MSO转换。加上我们最近的结果[LICS 2016],这意味着对于每1美元,都存在一个MSO转换,输入了树线性值$k$的图,非定值输出出其最佳宽度的树分解。我们还表明,MSO的移植可以在线性固定参数时间进行,使我们能够得出Bodlaender和Kloks的算法结果,作为我们主要结果的必然结果。