Uncrossed Multiflows and Applications to Disjoint Paths

A multiflow in a planar graph is uncrossed if its support paths do not cross. Recently such flows have 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 investigate the utility of uncrossed flow more generally and 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 some instances that have large clique demand graphs. Second, can we always round a fractional uncrossed flow to a "good" integral flow? We again answer positively. For maximization problems, we obtain integral flows with a constant fraction of the original value. For congestion problems (where we fully route all given demands), we obtain integral flows with edge congestion 2. Consequently, we obtain constant-factor approximation algorithms for maximum disjoint paths and minimum congestion integer multiflow for pairwise-planar instances, and show such instances have a constant integral flow-multicut gap. Finally, given a planar multiflow instance, can we determine if there exists a congestion-1 uncrossed fractional flow (congestion) or find the maximum value uncrossed fractional flow (maximization)? For congestion, we show this problem is NP-hard, but finding uncrossed edge-disjoint paths is polytime solvable if the demands span a bounded number of faces. For maximization, we present a strong inapproximability result.