Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other? We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider so-called α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}-orientations of a graph G, in which every vertex v has a specified outdegree α(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (v)$$\end{document}, and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}-orientations of a planar graph G is at most two is NP-complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope. We also consider the dual question of the flip distance between graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang et al. (Acta Math Sin Engl Ser 35(4):569–576, 2019).

中文翻译：

---

翻转图形方向之间的距离

翻转图是一类无处不在的图，它通过基本的局部变化对一组组合对象上的关系进行编码。例如，Associahedra 的骨架是由凸多边形三角剖分中的四边形翻转引起的图。对于翻转图的某些定义，要考虑的一个自然计算问题是翻转距离：给定两个对象，将一个对象转换为另一个对象所需的最少翻转次数是多少？我们在简单图的方向上考虑翻转图，其中翻转包括反转某些边的方向。更确切地说，我们证明决定两个α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \ usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} - 平面图 G 的方向最多为两个是 NP 完全的。这也适用于完美匹配的特殊情况，其中翻转涉及交替循环。这个问题相当于在两个分割拟阵的公共基多面体上，或者在一个凹形多面体上寻找测地线。因此，它提供了一个有趣的翻转距离问题示例，尽管在结构良好的组合多胞体上具有作为测地线的自然解释，但在计算上难以解决。我们还考虑了图形方向之间的翻转距离的双重问题，其中每个循环都有指定数量的前向边，而翻转是最小定向切割中所有边的反转。总的来说，问题仍然很棘手。但是，如果我们限制为只将汇变为源的翻转，反之亦然，那么问题可以在多项式时间内解决。在这里，我们利用翻转图是分布格的覆盖图这一事实。这概括了 Zhang 等人最近的结果。(Acta Math Sin Engl Ser 35(4):569–576, 2019)。如果我们限制为只将汇变为源的翻转，反之亦然，那么问题可以在多项式时间内解决。在这里，我们利用翻转图是分布格的覆盖图这一事实。这概括了 Zhang 等人最近的结果。(Acta Math Sin Engl Ser 35(4):569–576, 2019)。如果我们限制为只将汇变为源的翻转，反之亦然，那么问题可以在多项式时间内解决。在这里，我们利用翻转图是分布格的覆盖图这一事实。这概括了 Zhang 等人最近的结果。(Acta Math Sin Engl Ser 35(4):569–576, 2019)。