An elimination tree of a connected graph $G$ is a rooted tree on the vertices of $G$ obtained by choosing a root $v$ and recursing on the connected components of $G-v$ to obtain the subtrees of $v$. The graph associahedron of $G$ is a polytope whose vertices correspond to elimination trees of $G$ and whose edges correspond to tree rotations, a natural operation between elimination trees. These objects generalize associahedra, which correspond to the case where $G$ is a path. Ito et al. [ICALP 2023] recently proved that the problem of computing distances on graph associahedra is NP-hard. In this paper we prove that the problem, for a general graph $G$, is fixed-parameter tractable parameterized by the distance $k$. Prior to our work, only the case where $G$ is a path was known to be fixed-parameter tractable. To prove our result, we use a novel approach based on a marking scheme that restricts the search to a set of vertices whose size is bounded by a (large) function of $k$. On the negative side, we show that it is unlikely that FPT algorithms exist on a natural generalization of graph associahedra, namely hypergraphic polytopes, by proving that computing distances on them is W[2]-hard parameterized by the distance. We also prove that, on hypergraphic polytopes, the distance cannot be approximated in polynomial time within a factor $c \cdot \log(|V|+|\mathcal{E}|)$ for some constant $c > 0$ unless P = NP, where $H=(V, \mathcal{E})$ is the input hypergraph. This result strengthens the hardness result of Cardinal and Steiner [Combin. Theory 2025], who proved that the problem cannot be approximated within a factor $(1 + \varepsilon)$ for some absolute constant $\varepsilon > 0$ unless P = NP. Finally, we rule out the existence of polynomial kernels parameterized by the number of vertices of the input hypergraph, a parameter for which the problem is easily seen to be FPT.

翻译：连通图$G$的消去树是以$G$的顶点为节点的一棵有根树，其构造方式为：选取一个根顶点$v$，然后递归地对$G-v$的连通分量进行处理以得到$v$的子树。图$G$的图表多面体是一个多面体，其顶点对应于$G$的消去树，边对应于消去树之间的旋转操作（一种自然操作）。这些对象推广了通常的图表多面体，后者对应于$G$为路径的情形。Ito等人[ICALP 2023]近期证明了在图表多面体上计算距离的问题是NP困难的。本文证明：对于一般图$G$，该问题以距离$k$为参数具有固定参数易解性（FPT）。在此之前，仅$G$为路径的情形已知具有固定参数易解性。为证明这一结果，我们提出了一种基于标记方案的新方法，该方法将搜索限制在规模受$k的（大）函数限制的顶点集上。在否定方面，我们通过证明在图表多面体的自然推广——超图多面体上计算距离是W[2]困难的（以距离为参数），表明此类问题难以存在FPT算法。此外，我们证明了在超图多面体上，除非P=NP，否则无法在多项式时间内以因子$c \cdot \log(|V|+|\mathcal{E}|)$（其中$c>0$为常数，$H=(V,\mathcal{E})$为输入超图）近似距离。这一结果加强了Cardinal和Steiner[Combin. Theory 2025]的硬度结论，后者曾证明除非P=NP，否则无法在多项式时间内以因子$(1+\varepsilon)$（其中$\varepsilon>0$为绝对常数）近似该问题。最后，我们排除了以输入超图顶点数为参数的多项式核的存在性——该问题是关于此参数显然具有FPT性质的。