Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unsplittable Multicommodity Flows in Outerplanar Graphs

David Alemán-Espinosa, Nikhil Kumar

TL;DR

We address unsplittable multicommodity flows on outerplanar graphs under the cut-condition. The authors provide a polynomial-time scheme that yields a $(2\alpha_{RL}+1)$-feasible unsplittable flow, and since $\alpha_{RL} \le \tfrac{13}{10}$, this gives a $\frac{18}{5}$-feasible bound on edge-capacity violation. The approach uses a novel pinning technique with simultaneous, bounded capacity increases and reduces general outerplanar instances to ring-loading subproblems solved by a ring-loading oracle. For instances where all demands become good, a 1-feasible unsplittable flow is obtainable via a targeted outerplanar-to-planar decomposition, strengthening known results beyond single-face or cycle cases.

Abstract

We consider the problem of multicommodity flows in outerplanar graphs. Okamura and Seymour showed that the cut-condition is sufficient for routing demands in outerplanar graphs. We consider the unsplittable version of the problem and prove that if the cut-condition is satisfied, then we can route each demand along a single path by exceeding the capacity of an edge by no more than $\frac{18}{5} \cdot d_{max}$, where $d_{max}$ is the value of the maximum demand.

Unsplittable Multicommodity Flows in Outerplanar Graphs

TL;DR

We address unsplittable multicommodity flows on outerplanar graphs under the cut-condition. The authors provide a polynomial-time scheme that yields a -feasible unsplittable flow, and since , this gives a -feasible bound on edge-capacity violation. The approach uses a novel pinning technique with simultaneous, bounded capacity increases and reduces general outerplanar instances to ring-loading subproblems solved by a ring-loading oracle. For instances where all demands become good, a 1-feasible unsplittable flow is obtainable via a targeted outerplanar-to-planar decomposition, strengthening known results beyond single-face or cycle cases.

Abstract

We consider the problem of multicommodity flows in outerplanar graphs. Okamura and Seymour showed that the cut-condition is sufficient for routing demands in outerplanar graphs. We consider the unsplittable version of the problem and prove that if the cut-condition is satisfied, then we can route each demand along a single path by exceeding the capacity of an edge by no more than , where is the value of the maximum demand.
Paper Structure (19 sections, 14 theorems, 31 equations, 3 figures)

This paper contains 19 sections, 14 theorems, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Multicommodity Flows
  4. The cut-condition
  5. Planar Graphs
  6. The Ring Loading Problem
  7. Our Contributions
  8. Overview of Techniques
  9. Unsplittable Flows in Outerplanar Graphs
  10. Description of the Algorithm
  11. Overview:
  12. Description of an Iteration:
  13. Proof of Correctness
  14. Handling Parallel Edges in $G$
  15. Proof of Theorem \ref{['theorem:G+Hplanar']}
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $\mathcal{I}(G,c,D,d)$ be an instance of multicommodity flow such that $G$ is planar and all the edges of $H$ are incident on a fixed face i.e. there exists a face $f$ such that for each $(u,v) \in D$, both $u$ and $v$ lie on $f$, then the cut-condition is necessary and sufficient for the existe

Figures (3)

  • Figure 1: The figure on the left depicts an outerplanar graph $G$ where each edge $e\in E$ is labelled by $x^*(e)$, where $x^*$ is defined as in \ref{['eq:defx']}. The shaded region represents the face $f$ being considered during the iteration, and the yellow tree corresponds to the weak dual $G^*$ of $G$. The figure on the right depicts the resulting capacities $c_{\text{rl}}$ assigned to the ring loading instance $\mathcal{RL}([n],c_{\text{rl}},D^{(k)}_1\cup D^{(k)}_2,d^{(k)})$.
  • Figure 2: The algorithm terminates at the start of iteration 7. The demand edges of the resulting instance returned by the algorithm are $D^{(7)}=\mathcal{W}^{(7)}_b\cup\mathcal{W}^{(7)}_g$, where $\mathcal{W}^{(7)}_b=\{\textcolor{verde}{\{11,12\},\{12,13\},\{13,14\},\{14,15\},\{15,16\},\{16,20\},\{20,21\},\{21,22\}}\}$, $\mathcal{W}^{(7)}_g=\{\textcolor{morado}{\{9,5\},\{5,4\},\{4,3\},\{3,21\},\{21,20\},\{20,19\},\{19,18\}}\}$.
  • Figure :

Theorems & Definitions (32)

  • Theorem 2.1: Okamura-Seymour okamura1981multicommodity
  • Lemma 2.2: schrijver2003combinatorial
  • Theorem 2.3
  • Proposition 2.4
  • Theorem 3.0
  • Theorem 3.0
  • Lemma 4.1
  • proof
  • Theorem 4.1
  • Theorem 4.1
  • ...and 22 more