Node-capacitated ring routing

A. Frank, Bruce Shepherd, Vivek Tandon, Zoltán Végh

Research output: Contribution to journalArticle

7 Citations (Scopus)

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 languageEnglish
Pages (from-to)372-383
Number of pages12
JournalMathematics of Operations Research
Volume27
Issue number2
Publication statusPublished - May 2002

Fingerprint

Routing
Polynomials
Ring
Multicommodity Flow
Vertex of a graph
Strongly Polynomial Algorithm
Farkas Lemma
Ring Network
Integer
Node
Integrality
Routing Problem
Violate
Distance Function
Undirected Graph
Fractional
Non-negative
If and only if
Imply
Requirements

Keywords

  • Half-integral flow
  • Multicommodity flow
  • Ring routing

ASJC Scopus subject areas

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

Cite this

Frank, A., Shepherd, B., Tandon, V., & Végh, Z. (2002). Node-capacitated ring routing. Mathematics of Operations Research, 27(2), 372-383.

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

In: Mathematics of Operations Research, Vol. 27, No. 2, 05.2002, p. 372-383.

Research output: Contribution to journalArticle

Frank, A, Shepherd, B, Tandon, V & Végh, Z 2002, 'Node-capacitated ring routing', Mathematics of Operations Research, vol. 27, no. 2, pp. 372-383.
Frank A, Shepherd B, Tandon V, Végh Z. Node-capacitated ring routing. Mathematics of Operations Research. 2002 May;27(2):372-383.
Frank, A. ; Shepherd, Bruce ; Tandon, Vivek ; Végh, Zoltán. / Node-capacitated ring routing. In: Mathematics of Operations Research. 2002 ; Vol. 27, No. 2. pp. 372-383.
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