A multiflow in a planar graph is uncrossed if its support paths do not cross in the plane. 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.