# Stable Multicommodity Flows

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- each arc ${a}_{i}$ is unsaturated in f,
- (ii)
- ${v}_{1}=s$ or there is an arc ${a}^{\prime}={v}_{1}u$ for which $f\left({a}^{\prime}\right)>0$ and ${a}^{\prime}{<}_{{v}_{1}}{a}_{1}$,
- (iii)
- ${v}_{k}=t$ or there is an arc ${a}^{\u2033}=w{v}_{k}$ for which $f\left({a}^{\u2033}\right)>0$ and ${a}^{\u2033}{<}_{{v}_{k}}{a}_{k-1}$.

**Theorem**

**1.1**

**.**Every network with preferences has a stable flow. If the capacities are integral, then there is an integral stable flow.

#### 1.1. Stable Multicommodity Flows

- (i)
- each arc ${a}_{i}$ is unsaturated by ${f}^{j}$, that is, ${f}^{j}\left({a}_{i}\right)<{c}_{{a}_{i}}^{j}$,
- (ii)
- ${v}_{1}={s}^{j}$ or there is an arc ${a}^{\prime}={v}_{1}u$ for which ${f}^{j}\left({a}^{\prime}\right)>0$ and ${a}^{\prime}{<}_{{v}_{1}}^{j}{a}_{1}$,
- (iii)
- ${v}_{k}={t}^{j}$ or there is an arc ${a}^{\u2033}=w{v}_{k}$ for which ${f}^{j}\left({a}^{\u2033}\right)>0$ and ${a}^{\prime \prime}{<}_{{v}_{k}}^{j}{a}_{k-1}$,
- (iv)
- if an arc ${a}_{i}$ of P is saturated by f, i.e., ${\sum}_{j=1}^{\ell}{f}^{j}\left({a}_{i}\right)={c}_{{a}_{i}}$, then there is a commodity ${j}^{\prime}$ such that ${f}^{{j}^{\prime}}\left({a}_{i}\right)>0$ and ${j}^{\prime}{<}_{{a}_{i}}j$.

#### 1.2. A Polyhedral Version of Sperner’s Lemma

**Theorem**

**1.2**

**.**Let $P\subseteq {\mathbb{R}}^{n}$ be an n-dimensional pointed polyhedron whose characteristic cone is generated by n linearly independent vectors. If the facets of the polyhedron are coloured with n colours such that facets containing the i-th extreme direction do not get colour i, then there is a multicoloured vertex.

## 2. Existence of a Stable Multicommodity Flow

**Theorem**

**2.1.**

- We can define a suitable colouring of the facets of Π such that any multicoloured vertex of Π is on the face F,
- We show that a multicoloured vertex corresponds to a stable multicommodity flow. This is where the variables ${x}_{P}^{j}$ (where P has length at least 2) play a role: in some sense, they correspond to possibilities of changing a feasible solution along a blocking quasi-path.

- to an inequality of type (1) or type (2) we assign ${y}_{a}^{j}$,
- to an inequality of type (3) we assign ${x}_{P}^{j}$ for a longest possible quasi-paths $P\in {\mathcal{P}}^{\prime}$,
- to an inequality of type (4) we assign ${x}_{P}^{j}$ for a quasi-path $P\in {\mathcal{P}}^{\prime}$ in which the outgoing arc from v is smallest possible in the order ${<}_{v}^{j}$ from ${\mathcal{P}}^{\prime}$, and among these, we choose P to be one of the longest quasi-paths,
- to an inequality of type (5) we assign ${x}_{P}^{j}$ for a quasi-path $P\in {\mathcal{P}}^{\prime}$ in which the incoming arc to v is smallest possible in the order ${<}_{v}^{j}$ from ${\mathcal{P}}^{\prime}$, and among these, we choose P to be one of the longest quasi-paths,
- to an inequality of type (6) we assign ${x}_{P}^{j}$ where j is the commodity that is smallest in the order ${<}_{a}$ among those with nonempty ${\mathcal{P}}_{j}^{\prime}$, and from ${\mathcal{P}}_{j}^{\prime}$ we choose P to be one of the longest quasi-paths.

**Claim**

**2.2.**

**Claim**

**2.3.**

**Claim**

**2.4.**

**Claim**

**2.5.**

## 3. PPAD-Hardness

**Theorem**

**3.1.**

## 4. Open problems

## Acknowledgements

## References

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**MDPI and ACS Style**

Király, T.; Pap, J.
Stable Multicommodity Flows. *Algorithms* **2013**, *6*, 161-168.
https://doi.org/10.3390/a6010161

**AMA Style**

Király T, Pap J.
Stable Multicommodity Flows. *Algorithms*. 2013; 6(1):161-168.
https://doi.org/10.3390/a6010161

**Chicago/Turabian Style**

Király, Tamás, and Júlia Pap.
2013. "Stable Multicommodity Flows" *Algorithms* 6, no. 1: 161-168.
https://doi.org/10.3390/a6010161