A multiflow in a planar graph is uncrossed if its support paths do not cross. Recently such flows played a role in approximation algorithms for maximum disjoint paths in "fully-planar" instances, where the combined supply-demand graph is planar, as well as low-congestion unsplittable flows for fully-planar and single-source instances. We expand on the theory of uncrossed flows to investigate their utility more generally. We ask three key questions. First, are there other interesting planar multiflow instances that admit uncrossed flows? We answer affirmatively, demonstrating a new family of "pairwise-planar" instances whose flows can be uncrossed. This family subsumes fully-planar but includes substantially more, such as fully-compliant series-parallel instances, and has instances with clique demand graphs. Second, given a fractional uncrossed flow, can we always round it to a "good" integral flow? We again answer positively. For maximization problems (where we maximize the total amount of flow), we obtain integral flows with a constant fraction of the original value. For congestion problems (where we fully route specific given demands), we obtain integral flows with edge congestion 2, or unsplittable flows with an additional additive error. As a consequence we obtain approximation algorithms for maximum disjoint paths and minimum congestion integer multiflow for pairwise-planar instances. Finally, given a planar multiflow instance, can we determine if there exists a congestion 1 uncrossed fractional flow (congestion) or find the highest value uncrossed fractional flow (maximization)? For the congestion model, we show this is NP-hard, but finding uncrossed edge-disjoint paths is polytime solvable if the demands span a bounded number of faces. For the maximization model, we present a strong inapproximability result.

翻译：若平面图中多流的支撑路径互不交叉，则称该多流为非交叉流。近期，这类流在"全平面"实例（即供需组合图具有平面性）的最大不相交路径近似算法，以及全平面与单源实例的低拥塞不可分流算法中发挥了重要作用。本文拓展非交叉流理论以更广泛地探究其应用价值。我们提出三个核心问题：首先，是否存在其他可容纳非交叉流的有趣平面多流实例？我们给出肯定回答，通过构建新型"成对平面"实例族证明其流可被非交叉化。该实例族包含全平面情形，但扩展范围更广，例如包含完全合规的串并联实例，且能构造具有团需求图的实例。其次，给定分数非交叉流，是否总能将其舍入为"优质"整数流？我们再次给出肯定答案。对于最大化问题（以总流量最大化为目标），我们获得具有原值常数比例的整数流；对于拥塞问题（需完全路由特定给定需求），我们获得边拥塞为2的整数流，或具有附加加性误差的不可分流。由此我们为成对平面实例的最大不相交路径和最小拥塞整数多流问题提出了近似算法。最后，给定平面多流实例，能否判定是否存在拥塞为1的非交叉分数流（拥塞模型），或找到最高值的非交叉分数流（最大化模型）？针对拥塞模型，我们证明该判定问题是NP难的，但当需求跨越有限个面时，寻找非交叉边不相交路径可在多项式时间内求解。针对最大化模型，我们提出了强不可近似性结果。