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An Approximate Generalization of the Okamura-Seymour Theorem

Nikhil Kumar

TL;DR

This work extends the Okamura-Seymour theorem to planar graphs by proving an $O(1)$ flow-cut gap for demand pairs whose endpoints lie on a common face, even when demands are spread across multiple faces. It achieves this via a metric-embedding approach: constructing an $L_1$ embedding of the planar metric that preserves distances for face-local demand pairs within a constant factor, and combining several embedding schemes (geodesic pairs, single-source paths, and constrained extensions) through a laminar structure of face supports. The core contributions are the laminar-face framework, the alpha-good length decomposition, and the constrained-extension lemmas, which together enable a global $L_1$ embedding with constant distortion and thus a constant-factor flow routing guarantee from the cut-condition. This yields a substantial step toward constant flow-cut gaps in planar graphs, with implications for broader classes and potential paths toward the GNRS conjecture in this setting.

Abstract

We consider the problem of multicommodity flows in planar graphs. Okamura and Seymour showed that if all the demands are incident on one face, then the cut-condition is sufficient for routing demands. We consider the following generalization of this setting and prove an approximate max flow-min cut theorem: for every demand edge, there exists a face containing both its end points. We show that the cut-condition is sufficient for routing $Ω(1)$-fraction of all the demands. To prove this, we give a $L_1$-embedding of the planar metric which approximately preserves distance between all pair of points on the same face.

An Approximate Generalization of the Okamura-Seymour Theorem

TL;DR

This work extends the Okamura-Seymour theorem to planar graphs by proving an flow-cut gap for demand pairs whose endpoints lie on a common face, even when demands are spread across multiple faces. It achieves this via a metric-embedding approach: constructing an embedding of the planar metric that preserves distances for face-local demand pairs within a constant factor, and combining several embedding schemes (geodesic pairs, single-source paths, and constrained extensions) through a laminar structure of face supports. The core contributions are the laminar-face framework, the alpha-good length decomposition, and the constrained-extension lemmas, which together enable a global embedding with constant distortion and thus a constant-factor flow routing guarantee from the cut-condition. This yields a substantial step toward constant flow-cut gaps in planar graphs, with implications for broader classes and potential paths toward the GNRS conjecture in this setting.

Abstract

We consider the problem of multicommodity flows in planar graphs. Okamura and Seymour showed that if all the demands are incident on one face, then the cut-condition is sufficient for routing demands. We consider the following generalization of this setting and prove an approximate max flow-min cut theorem: for every demand edge, there exists a face containing both its end points. We show that the cut-condition is sufficient for routing -fraction of all the demands. To prove this, we give a -embedding of the planar metric which approximately preserves distance between all pair of points on the same face.
Paper Structure (12 sections, 20 theorems, 11 equations, 4 figures)

This paper contains 12 sections, 20 theorems, 11 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Embedding Metrics into $L_1$
  4. Flow-Cut Gap and Embedding into $L_1$
  5. Cut Metrics and $L_1$ Embedding
  6. Our Contribution
  7. Laminar Structure of Face Supports
  8. Embedding For The Geodesic Pairs
  9. Single Source Shortest Path Embeddings
  10. Constrained Embedding
  11. Putting Everything Together
  12. Conclusions

Key Result

Theorem 1

If $G$ is a planar graph, all the edges of $H$ are restricted to a face and $G+H$ is Eulerian, then the cut-condition is necessary and sufficient for integral routing of all the demands.

Figures (4)

  • Figure 1: This example first appeared in the work of Okamura and Seymour okamura1981multicommodity. All supply (solid) and demand (dashed) edges have value 1. $S$ (bold-dashed) is a cut. The total capacity of supply edges across $S$ is three while $S$ separates three units of demand; hence $S$ satisfies the cut-condition. One can check that no cut violates the cut-condition. Since the source-sink of every demand is distance two apart, a total capacity of $4 \cdot 2=8$ is required for a feasible routing, but only six are available. Hence, no feasible routing is possible. This also implies that no more than 3/4 of every demand can be routed simultaneously. The figure on the right shows a feasible routing of 3/4 of every demand, which implies a flow-cut gap of 4/3.
  • Figure 2: Illustrations for the proof of Lemma \ref{['lemma:laminar rfuv']}.
  • Figure 3: Illustration for the statement of Lemma \ref{['Lemma: face_support_laminar_final']}.
  • Figure 4: The figure shows a face $f$ and the terms associated with it in Theorem \ref{['Theorem: Separable Instances']}. $X_1,Y_1,X_2,Y_2,X_3,Y_3$ form a partition of the vertex set of face $f$ and $R_f$ consists of $d_1,d_2,d_3,d_4,d_5$.

Theorems & Definitions (32)

  • Theorem 1: Okamura -Seymour okamura1981multicommodity
  • Theorem 2: Seymour seymour1981odd
  • Lemma 1: schrijver2003combinatorial
  • Theorem 3: Okamura-Seymourokamura1981multicommodity
  • Theorem 4: Seymourseymour1981odd
  • Theorem 5
  • Theorem 6
  • Claim 1
  • Claim 2
  • Lemma 2
  • ...and 22 more