Uncrossed Multiflows and Applications to Disjoint Paths

2025-10-31

AI Summary

This paper initiates the systematic study of uncrossed multiflows in planar graphs as a standalone algorithmic problem. For general (not necessarily fully) planar graphs, it formulates two variants: congestion minimization—satisfying all demands under edge- or vertex-capacity constraints—and flow maximization—maximizing the total integer flow value subject to supply constraints. The approach combines topological planar graph analysis, fractional flow construction, and a novel integral rounding technique to achieve low-congestion or high-value integer solutions. Key theoretical contributions include: proving NP-hardness of congestion feasibility, while showing polynomial-time solvability when the number of demand faces is bounded; establishing strong inapproximability for the maximization variant; and deriving the first tight upper bound on the integrality gap for general planar graphs—generalizing prior results for fully planar instances—and providing a constant-factor integer rounding guarantee.

Technology Category

• Planning, Routing, and Scheduling: Planning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• Finding uncrossed multiflows in general planar graphs
• Studying NP-hardness and polynomial solvability in congestion models
• Analyzing integrality gaps and rounding approximations for flows

Innovation

• Developed uncrossed multiflow algorithms for general planar graphs
• Proved NP-hardness for congestion model with bounded faces
• Established constant-factor rounding for integral multiflow maximization