Uncrossed Multiflows and Applications to Disjoint Paths

TL;DR

This work investigates uncrossed multiflows in planar graphs under two models—maximization and congestion—allowing fractional and integral solutions and introducing U-constrained variants. It provides a spectrum of results: NP-hardness for deciding the existence of a fractional uncrossed multiflow routing all demands, polynomial-time solvability for integral uncrossed flows when demands touch a bounded number of faces, and strong inapproximability for the maximization setting. For maximization, it develops an approximate integer decomposition via uncrossed string graphs to obtain constant-factor rounding and ties these results to reductions from capped MEDP/MNDP, establishing strong hardness. For congestion minimization, it presents a rounding scheme from strongly uncrossed fractional solutions to integral flows with edge congestion at most $2$, and with additive tolerance $d_{ ext{max}}$ achieves unsplittable routings; sector-based arguments underpin these guarantees. Finally, it reduces integral uncrossed flow to node-disjoint instances through degree-reduction gadgets, connecting uncrossed flow feasibility to classical planarity tools and highlighting the broader applicability of uncrossed techniques to pairwise-planar and series-compliant classes.

Abstract

A multiflow in a planar graph is uncrossed if the curves identified by its support paths do not cross in the plane. Such flows have played a role in previous routing algorithms, including Schrijver's Homotopy Method and unsplittable flows in directed planar single-source instances. Recently uncrossed flows have played a key role in approximation algorithms for maximum disjoint paths in fully-planar instances, where the combined supply plus demand graph is planar. In the fully-planar case, any fractional multiflow can be converted into one that is uncrossed, which is then exploited to find a good rounding of the fractional solution. We investigate finding an uncrossed multiflow as a standalone algorithmic problem in general planar instances (not necessarily fully-planar). We consider both a congestion model where the given demands must all be routed, and a maximization model where the goal is to pack as much flow in the supply graph as possible (not necessarily equitably). For the congestion model, we show that determining if an instance has an uncrossed (fractional) multiflow is NP-hard, but the problem of finding an integral uncrossed flow is polytime solvable if the demands span a bounded number of faces. For the maximization model, we present a strong (almost polynomial) inapproximability result. Regarding integrality gaps, for maximization we show that an uncrossed multiflow in a planar instance can always be rounded to an integral multiflow with a constant fraction of the original value. This holds in both the edge-capacitated and node-capacitated settings, and generalizes earlier bounds for fully-planar instances. In the congestion model, given an uncrossed fractional multiflow, we give a rounding procedure that produces an integral multiflow with edge congestion 2, which can be made unsplittable with an additional additive error of the maximum demand.

Figures (13)

• Figure 1: A fully-planar instance with two different multiflows routing its demands. Dashed edges represent demands, and coloured paths are flow paths. The flows paths on the left cross, while the multiflow on the right is uncrossed.
• Figure 2: (Left) Two edge-disjoint paths that cross at a common node. (Center) Two paths that cross at a common subpath. (Right) Two paths that share a common subpath, but are uncrossed. Uncoloured edges are shared between the two paths.
• Figure 3: (Left) An uncrossed parallelization of the uncrossed paths from \ref{['fig:crossing']}. (Right) Disk-expansion of the node $z$ after uncrossed parallelization.
• Figure 4: A very simple pairwise-planar instance. Each $G_i$ is some planar graph. The demand edges cannot be simultaneously embedded so that $G+H$ is planar ($G+H$ contains a $K_{3, 3}$-minor assuming each $G_i$ is connected). However, any pair of demands can be drawn without crossing by putting one in the inside face, and one on the outside face. Of course, one could make the instance more complicated by putting demands inside the $G_i$'s following a similar pattern.
• Figure 5: Problematic instances for uncrossability. (Left) A congestion instance with a feasible multiflow but no uncrossed multiflow routing both demands. (Right) The Keyhole Instance, a congestion instance with a feasible uncrossed trail-multiflow but no uncrossed path-multiflow.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1: Crossing
• Definition 2.2: Uncrossed Multiflow
• Definition 2.3: Quasicrossing
• Definition 2.4: Strongly Uncrossed
• Definition 3.1: Uncrossed String Graph, Realization