共签交叉族与外平面装置
Cosigning Crossing Families and Outer-Planar Gadgets
AI总结：
研究共签交叉族与外平面装置，通过算法性条件确定其存在性，并在平面图中用于寻找不相交边割集。
摘要：
设F是定义在顶点集V上的交叉族，即对于任何两个集合U,W∈F，若它们有非空交集且有真并集，则U∩W和U∪W也都在F中。令σ:V→{+,-}为一个签。我们称σ为“共签”如果每个集合包含一个正元素并排除一个负元素。它被称为“∩∪-闭”如果任何两个非空交集和共交集分别包含正负元素。我们通过必要且充分条件来刻画（∩∪-闭）共签σ的存在性。我们的证明是算法性的，并导致寻找σ的优雅“强制”算法，类似于Cameron-Edmonds算法用于二色化平衡超图。我们证明这些算法在多项式时间内运行，并进一步，通过子模函数最小化应用，共签算法可以通过oracle多项式时间运行。共签交叉族自然出现在具有顶点集V的有向图中，其中V由源和汇组成，其中F中的每个集合都由一条入边覆盖。在温和且必要的条件下，当我们把顶点放在圆周上时，我们可以构建F的外平面弧覆盖。这些装置随后用于在0,1加权平面有向图中寻找不相交的边割集，当权重为1的边形成一个连通组件但不一定是满连通时。
英文摘要：
Let $F$ be a crossing family over ground set $V$, that is, for any two sets $U,W\in{F}$ with nonempty intersection and proper union, both sets $U\cap{W},U\cup{W}$ are in $F$. Let $σ:V\to \{+,-\}$ be a signing. We call $σ$ a "cosigning" if every set includes a positive element and excludes a negative element. It is "$\cap\cup$-closed" if every pairwise nonempty intersection and co-intersection include positive and negative elements, respectively. We characterize the existence of ($\cap\cup$-closed) cosignings $σ$ through necessary and sufficient conditions. Our proofs are algorithmic and lead to elegant `forcing' algorithms for finding $σ$, reminiscent of the Cameron-Edmonds algorithm for bicoloring balanced hypergraphs. We prove that the algorithms run in polynomial time, and further, the cosigning algorithm can be run in oracle polynomial time through an application of submodular function minimization. Cosigned crossing families arise naturally in digraphs with vertex set $V$ comprised of sources and sinks, where every set in $F$ is "covered" by an incoming arc. Under mild and necessary conditions, we build an outer-planar arc covering of $F$ when the vertices are placed around a circle. These gadgets are then used to find disjoint dijoins in $0,1$-weighted planar digraphs when the weight-$1$ arcs form a connected component that is not necessarily spanning.
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