Base polytopes of polymatroids, also known as generalized permutohedra, are polytopes whose edges are parallel to a vector of the form $\mathbf{e}_i - \mathbf{e}_j$. We consider the following computational problem: Given two vertices of a generalized permutohedron $P$, find a shortest path between them on the skeleton of $P$. This captures many known flip distance problems, such as computing the minimum number of exchanges between two spanning trees of a graph, the rotation distance between binary search trees, the flip distance between acyclic orientations of a graph, or rectangulations of a square. We prove that this problem is $NP$-hard, even when restricted to very simple polymatroids in $\mathbb{R}^n$ defined by $O(n)$ inequalities. Assuming $P\not= NP$, this rules out the existence of an efficient simplex pivoting rule that performs a minimum number of nondegenerate pivoting steps to an optimal solution of a linear program, even when the latter defines a polymatroid. We also prove that the shortest path problem is inapproximable when the polymatroid is specified via an evaluation oracle for a corresponding submodular function, strengthening a recent result by Ito et al. (ICALP'23). More precisely, we prove the $APX$-hardness of the shortest path problem when the polymatroid is a hypergraphic polytope, whose vertices are in bijection with acyclic orientations of a given hypergraph. The shortest path problem then amounts to computing the flip distance between two acyclic orientations of a hypergraph. On the positive side, we provide a polynomial-time approximation algorithm for the problem of computing the flip distance between two acyclic orientations of a hypergraph, where the approximation factor is the maximum codegree of the hypergraph. Our result implies an exact polynomial-time algorithm for the flip distance between two acyclic orientations of any linear hypergraph.

翻译：多拟阵的基多面体（也称为广义置换体）是指边平行于形式为 $\mathbf{e}_i - \mathbf{e}_j$ 的向量的多面体。我们考虑以下计算问题：给定广义置换体 $P$ 的两个顶点，在 $P$ 的骨架（即边图）上找到它们之间的最短路径。这一问题涵盖了许多已知的翻转距离问题，例如计算图的两棵生成树之间的最小交换次数、二叉搜索树之间的旋转距离、图的无环定向之间的翻转距离，以及正方形的矩形剖分之间的翻转距离。我们证明，即使限制在由 $O(n)$ 个不等式定义的 $\mathbb{R}^n$ 中的极简单多拟阵上，该问题也是 $NP$-难的。假设 $P \neq NP$，这排除了存在一种高效单纯形转轴规则的可能性，该规则能以最少的非退化转轴步骤到达线性规划的最优解，即使该线性规划定义了一个多拟阵。我们还证明，当多拟阵通过相应子模函数的求值预言机（oracle）指定时，最短路径问题是不可近似逼近的，这强化了 Ito 等人（ICALP'23）最近的结果。更精确地说，我们证明了当多拟阵为超图多面体时，最短路径问题是 $APX$-难的；超图多面体的顶点与给定超图的无环定向一一对应。此时，最短路径问题等价于计算超图的两个无环定向之间的翻转距离。在积极方面，我们为计算超图的两个无环定向之间的翻转距离问题提供了一个多项式时间近似算法，其近似比等于超图的最大代码度。我们的结果蕴涵了任意线性超图的两个无环定向之间翻转距离的多项式时间精确算法。