We devise constant-factor approximation algorithms for finding as many disjoint cycles as possible from a certain family of cycles in a given planar or bounded-genus graph. Here disjoint can mean vertex-disjoint or edge-disjoint, and the graph can be undirected or directed. The family of cycles under consideration must satisfy two properties: it must be uncrossable and allow for an oracle access that finds a weight-minimal cycle in that family for given nonnegative edge weights or (in planar graphs) the union of all remaining cycles in that family after deleting a given subset of edges. Our setting generalizes many problems that were studied separately in the past. For example, three families that satisfy the above properties are (i) all cycles in a directed or undirected graph, (ii) all odd cycles in an undirected graph, and (iii) all cycles in an undirected graph that contain precisely one demand edge, where the demand edges form a subset of the edge set. The latter family (iii) corresponds to the classical disjoint paths problem in fully planar and bounded-genus instances. While constant-factor approximation algorithms were known for edge-disjoint paths in such instances, we improve the constant in the planar case and obtain the first such algorithms for vertex-disjoint paths. We also obtain approximate min-max theorems of the Erd\H{o}s--P\'osa type. For example, the minimum feedback vertex set in a planar digraph is at most 12 times the maximum number of vertex-disjoint cycles.

翻译：我们设计了常数因子近似算法，用于在给定的平面图或有界亏格图中，从某类圈族中找出尽可能多的不相交圈。这里的不相交可指顶点不相交或边不相交，且图可以是有向或无向的。所考虑的圈族需满足两个性质：它是不可交叉的，并且允许通过预言机访问，在给定非负边权时找到该族中权重最小的圈，或（在平面图中）在删除给定边子集后找到该族中所有剩余圈的并集。我们的设定概括了过去分别研究的许多问题。例如，满足上述性质的三个圈族包括：(i) 有向图或无向图中的所有圈，(ii) 无向图中的所有奇圈，以及(iii) 无向图中恰好包含一条需求边的所有圈，其中需求边构成边集的一个子集。最后一个圈族 (iii) 对应于全平面和有界亏格实例中的经典不交路径问题。虽然此前这些实例中已知边不交路径的常数因子近似算法，但我们改进了平面情形中的常数，并首次获得了顶点不交路径的算法。我们还得到了 Erdős–Pósa 类型的近似极小-极大定理。例如，平面有向图中的最小反馈顶点集至多是顶点不交圈最大数目的12倍。