We consider the $$\#\hbox {P}$$#P-complete problem of counting the number of linear extensions of a poset $$(\textsc {\#LE})$$(#LE); a fundamental problem in order theory with applications in a variety of distinct areas. In particular, we study the complexity of $$\textsc {\#LE}$$#LE parameterized by the well-known decompositional parameter treewidth for two natural graphical representations of the input poset, i.e., the cover and the incomparability graph. Our main result shows that $$\textsc {\#LE}$$#LE is fixed-parameter intractable parameterized by the treewidth of the cover graph. This resolves an open problem recently posed in the Dagstuhl seminar on Exact Algorithms. On the positive side we show that $${\textsc {\#LE}}$$#LE becomes fixed-parameter tractable parameterized by the treewidth of the incomparability graph.

中文翻译：

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计算线性扩展：树宽参数化

我们考虑 $$\#\hbox {P}$$#P-complete 问题，即计算偏序集的线性扩展数 $$(\textsc {\#LE})$$(#LE); 序理论中的一个基本问题，可应用于各种不同的领域。特别是，我们研究了 $$\textsc {\#LE}$$#LE 的复杂性，该复杂性由众所周知的分解参数 treewidth 参数化，用于输入偏序集的两个自然图形表示，即覆盖图和不可比性图。我们的主要结果表明 $$\textsc {\#LE}$$#LE 是由覆盖图的树宽参数化的固定参数难以处理的。这解决了最近在关于精确算法的 Dagstuhl 研讨会上提出的一个悬而未决的问题。从积极的方面来看，我们表明 $${\textsc {\#LE}}$$#LE 成为由不可比图的树宽参数化的固定参数易处理。