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Uncrossed Multiflows and Applications to Disjoint Paths

Authors Chandra Chekuri , Guyslain Naves , Joseph Poremba , F. Bruce Shepherd

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Network flows
  • Mathematics of computing → Approximation algorithms
Keywords
  • Network Flows
  • Disjoint Paths
  • Planar Graphs
  • Crossing
  • Flow-Multicut Gap
  • Approximation Algorithms
  • Combinatorial Optimization

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Abstract

A multiflow in a planar graph is uncrossed if the curves identified by its support paths do not cross in the plane. Recently, uncrossed flows have played a role in approximation algorithms for maximum disjoint paths in "fully-planar" instances, where the combined supply-plus-demand graph is planar. They are also used in algorithms to find low-congestion unsplittable flows for both fully-planar and single-source instances. For these two instance classes, any fractional multiflow can be converted into one that is uncrossed, which these algorithms then exploit to obtain their results.
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 (beyond fully-planar and single-source)? We answer affirmatively, demonstrating a new family of "pairwise-planar" instances whose fractional flows can be uncrossed. This family subsumes fully-planar but includes substantially more, such as (2-connected) 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 show any fractional uncrossed flow can be rounded to an integral flow with a constant fraction of its value. For congestion problems (where we must fully route all given demands), we give a rounding procedure that yields an integral multiflow 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 we ask, given an arbitrary planar instance, can we determine if there exists a congestion-1 uncrossed fractional flow (congestion setting) or find the maximum value uncrossed fractional flow (maximization setting)? 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 (almost-polynomial) inapproximability result.

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Chandra Chekuri, Guyslain Naves, Joseph Poremba, and F. Bruce Shepherd. Uncrossed Multiflows and Applications to Disjoint Paths. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 12:1-12:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.12

Author Details

Chandra Chekuri
  • Department of Computer Science, University of Illinois, Urbana-Champaign, IL, USA
Guyslain Naves
  • Aix-Marseille Univ, CNRS, LIS, Marseille, France
Joseph Poremba
  • Department of Computer Science, University of British Columbia, Vancouver, Canada
F. Bruce Shepherd
  • Department of Computer Science, University of British Columbia, Vancouver, Canada

Funding

  • Chekuri, Chandra: Supported in part by NSF grant CCF-2402667.
  • Shepherd, F. Bruce: Supported by NSERC Discovery Grant RGPIN-2026-06972.

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