### Abstract

We consider the node-capacitated routing problem in an undirected ring network along with its fractional relaxation, the node-capacitated multicommodity, flow problem. For the feasibility problem. Farkas' lemma provides a characterization for general undirected graphs, asserting roughly that there exists such a flow if and only if the so-called distance inequality holds for every choice of distance functions arising from nonnegative node weights. For rings, this (straightforward) result will be improved in two ways. We prove that, independent of the integrality of node capacities, it suffices to require the distance inequality only for distances arising from (0-1-2)-valued node weights, a requirement that will be called the double-cut conditions. Moreover, for integer-valued node capacities, the double-cut condition implies the existence of a half-integral multicommodity flow. In this case there is even an integer-valued multicommodity flow that violates each node capacity by at most one. Our approach gives rise to a combinatorial, strongly polynomial algorithm to compute either a violating double-cut or a node-capacitated multicommodity flow. A relation of the problem to its edge-capacitated counterpart will also be explained.

Original language | English |
---|---|

Pages (from-to) | 372-383 |

Number of pages | 12 |

Journal | Mathematics of Operations Research |

Volume | 27 |

Issue number | 2 |

Publication status | Published - May 2002 |

### Fingerprint

### Keywords

- Half-integral flow
- Multicommodity flow
- Ring routing

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Management Science and Operations Research

### Cite this

*Mathematics of Operations Research*,

*27*(2), 372-383.

**Node-capacitated ring routing.** / Frank, A.; Shepherd, Bruce; Tandon, Vivek; Végh, Zoltán.

Research output: Contribution to journal › Article

*Mathematics of Operations Research*, vol. 27, no. 2, pp. 372-383.

}

TY - JOUR

T1 - Node-capacitated ring routing

AU - Frank, A.

AU - Shepherd, Bruce

AU - Tandon, Vivek

AU - Végh, Zoltán

PY - 2002/5

Y1 - 2002/5

N2 - We consider the node-capacitated routing problem in an undirected ring network along with its fractional relaxation, the node-capacitated multicommodity, flow problem. For the feasibility problem. Farkas' lemma provides a characterization for general undirected graphs, asserting roughly that there exists such a flow if and only if the so-called distance inequality holds for every choice of distance functions arising from nonnegative node weights. For rings, this (straightforward) result will be improved in two ways. We prove that, independent of the integrality of node capacities, it suffices to require the distance inequality only for distances arising from (0-1-2)-valued node weights, a requirement that will be called the double-cut conditions. Moreover, for integer-valued node capacities, the double-cut condition implies the existence of a half-integral multicommodity flow. In this case there is even an integer-valued multicommodity flow that violates each node capacity by at most one. Our approach gives rise to a combinatorial, strongly polynomial algorithm to compute either a violating double-cut or a node-capacitated multicommodity flow. A relation of the problem to its edge-capacitated counterpart will also be explained.

AB - We consider the node-capacitated routing problem in an undirected ring network along with its fractional relaxation, the node-capacitated multicommodity, flow problem. For the feasibility problem. Farkas' lemma provides a characterization for general undirected graphs, asserting roughly that there exists such a flow if and only if the so-called distance inequality holds for every choice of distance functions arising from nonnegative node weights. For rings, this (straightforward) result will be improved in two ways. We prove that, independent of the integrality of node capacities, it suffices to require the distance inequality only for distances arising from (0-1-2)-valued node weights, a requirement that will be called the double-cut conditions. Moreover, for integer-valued node capacities, the double-cut condition implies the existence of a half-integral multicommodity flow. In this case there is even an integer-valued multicommodity flow that violates each node capacity by at most one. Our approach gives rise to a combinatorial, strongly polynomial algorithm to compute either a violating double-cut or a node-capacitated multicommodity flow. A relation of the problem to its edge-capacitated counterpart will also be explained.

KW - Half-integral flow

KW - Multicommodity flow

KW - Ring routing

UR - http://www.scopus.com/inward/record.url?scp=0036577325&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036577325&partnerID=8YFLogxK

M3 - Article

VL - 27

SP - 372

EP - 383

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

SN - 0364-765X

IS - 2

ER -