Let \(D=(V,A)\) be a digraph. For an integer \(k\ge 1\), a k-arc-connected flip is an arc subset of D such that after reversing the arcs in it the digraph becomes (strongly) k-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a k-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer \(\tau \ge 1\), suppose \(d_A^+(U)+(\frac{\tau }{k}-1)d_A^-(U)\ge \tau \) for all \(U\subsetneq V, U\ne \emptyset \), where \(d_A^+(U)\) and \(d_A^-(U)\) denote the number of arcs in A leaving and entering U, respectively. Let \({\mathcal {C}}\) be a crossing family over ground set V, and let \(f:{\mathcal {C}}\rightarrow {\mathbb {Z}}\) be a crossing submodular function such that \(f(U)\ge \frac{k}{\tau }(d_A^+(U)-d_A^-(U))\) for all \(U\in {\mathcal {C}}\). Then D has a k-arc-connected flip J such that \(f(U)\ge d_J^+(U)-d_J^-(U)\) for all \(U\in {\mathcal {C}}\). The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams’ so-called weak orientation theorem, and proves a weaker variant of Woodall’s conjecture on digraphs whose underlying undirected graph is \(\tau \)-edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral.

设 \(D=(V,A)\) 为有向图。对于整数 \(k\ge 1\)，k 弧连接翻转是 D 的弧子集，这样在反转其中的弧之后，有向图就变成（强）k 弧连接。本文的第一个主要结果介绍了 k 弧连接翻转存在的充分条件，该翻转也是交叉子模函数的子模流。更具体地说，给定一些整数 \(\tau \ge 1\)，假设 \(d_A^+(U)+(\frac{\tau }{k}-1)d_A^-(U)\ge \tau \ ) 对于所有 \(U\subsetneq V, U\ne \emptyset \)，其中 \(d_A^+(U)\) 和 \(d_A^-(U)\) 表示 A 中离开和进入的弧数分别为U。设 \({\mathcal {C}}\) 为地面集 V 上的交叉族，设 \(f:{\mathcal {C}}\rightarrow {\mathbb {Z}}\) 为交叉子模函数使得 \(f(U)\ge \frac{k}{\tau }(d_A^+(U)-d_A^-(U))\) 对于所有 \(U\in {\mathcal {C}} \）。那么 D 有一个 k 弧连接的翻转 J，使得对于所有 \(U\in {\mathcal {C}} 来说 \(f(U)\ge d_J^+(U)-d_J^-(U)\) \）。结果在图方向和组合优化中有多种应用。特别是，它强化了纳什-威廉姆斯所谓的弱方向定理，并证明了伍德尔关于有向图的猜想的较弱变体，该有向图的底层无向图是 \(\tau \)-边连通的。本文的第二个主要结果更加具有普遍性。它引入了两个子模流系统交集容量积分解存在的充分条件。这个充分条件暗示了 Edmonds 和 Giles 关于子模流系统的盒全对偶完整性的经典结果。它还具有这样的结果：在弱连通有向图中，两个子模流系统的交集是完全对偶积分的。