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:我们设计了恒定因素近似算法, 以在给定的平面图或捆绑式genus图中从某个周期的组合中找到尽可能多的脱节周期。 这里的脱节可以意味着顶点分解或边缘分解, 图形也可以不定向或定向。 所考虑周期的组合必须满足两个属性 : 它必须是不可交叉的, 并且允许在一个家庭中找到一个重量- 最小周期, 在给定的非负边缘重量或( 在平面图中) 所有剩余周期在删除给定的边缘子组之后的组合。 我们的设置将过去分别研究的许多问题概括化。 例如, 满足上述属性的三个家族是 (一) 定向或非定向图形中的所有循环, (二) 所有奇怪的循环都必须满足两种特性: 它必须是不可交叉的, 并且允许一个非定向的图表中包含一个准确的需求边缘, 需求边缘构成一个子群。 后一个家族(三) 在完全平面图中, 最典型的脱节路径中, 最明显的平面图是我们所知道的平面平面平面图中, 的平面图中, 最常态中, 最常数的平流- 。 同时, 在常态中, 我们的平面图中, 最常变的轨道中, 的轨道中, 也会改进的轨道- 例中, 在常态- 例中, 在常态- 。