一句话：证明宽度 k 树分解到最优宽度树分解可由 MSO 转换实现，并由此导出线性 FPT 算法。
论文研究如何用一元二阶逻辑（MSO）表达树分解的优化。经典 Bodlaender–Kloks 算法可在固定参数线性时间内，由宽度 k 的树分解计算最优宽度树分解。本文证明对每个正整数 k，存在从宽度 k 树分解到最优宽度树分解的 MSO 转换；结合作者先前 LICS 2016 的结果，可得对每个 k 存在 MSO 转换，输入树宽 k 的图，非确定性地输出其最优宽度树分解。作者还证明 MSO 转换可在固定参数线性时间内实现，从而把 Bodlaender–Kloks 的算法结果作为主定理的推论导出。
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.