Combinatorial algorithms for the unsplittable flow problem

Yossi Azar, Oded Regev

Research output: Contribution to journalArticle

Abstract

We provide combinatorial algorithms for the unsplittable flow problem (UFP) that either match or improve the previously best results. In the UFP we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path subject to the capacity constraints such that the total profit of the connected pairs is maximized.We consider three variants of the problem. First is the classical UFP in which the maximum demand is at most the minimum edge capacity. It was previously known to have an O(√m) approximation algorithm; the algorithm is based on the randomized rounding technique and its analysis makes use of the Chernoff bound and the FKG inequality.We provide a combinatorial algorithm that achieves the same approximation ratio and whose analysis is considerably simpler. Second is the extended UFP in which some demands might be higher than edge capacities. Our algorithm for this case improves the best known approximation ratio. We also give a lower bound that shows that the extended UFP is provably harder than the classical UFP. Finally, we consider the bounded UFP in which the maximum demand is at most 1/K times the minimum edge capacity for some K > 1. Here we provide combinatorial algorithms that match the currently best known algorithms. All of our algorithms are strongly polynomial and some can even be used in the online setting.

Original languageEnglish (US)
Pages (from-to)49-66
Number of pages18
JournalAlgorithmica (New York)
Volume44
Issue number1
DOIs
StatePublished - Jan 2006

Fingerprint

Combinatorial Algorithms
Profitability
Profit
FKG Inequality
Randomized Rounding
Directed graphs
Approximation algorithms
Capacity Constraints
Best Approximation
Directed Graph
Approximation Algorithms
Polynomials
Lower bound
Path
Polynomial
Subset
Approximation

Keywords

  • Combinatorial algorithms
  • Unsplittable flow problem

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design
  • Software
  • Safety, Risk, Reliability and Quality
  • Applied Mathematics

Cite this

Combinatorial algorithms for the unsplittable flow problem. / Azar, Yossi; Regev, Oded.

In: Algorithmica (New York), Vol. 44, No. 1, 01.2006, p. 49-66.

Research output: Contribution to journalArticle

@article{2349bc01d3b74645b0ac9093611c42f3,
title = "Combinatorial algorithms for the unsplittable flow problem",
abstract = "We provide combinatorial algorithms for the unsplittable flow problem (UFP) that either match or improve the previously best results. In the UFP we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path subject to the capacity constraints such that the total profit of the connected pairs is maximized.We consider three variants of the problem. First is the classical UFP in which the maximum demand is at most the minimum edge capacity. It was previously known to have an O(√m) approximation algorithm; the algorithm is based on the randomized rounding technique and its analysis makes use of the Chernoff bound and the FKG inequality.We provide a combinatorial algorithm that achieves the same approximation ratio and whose analysis is considerably simpler. Second is the extended UFP in which some demands might be higher than edge capacities. Our algorithm for this case improves the best known approximation ratio. We also give a lower bound that shows that the extended UFP is provably harder than the classical UFP. Finally, we consider the bounded UFP in which the maximum demand is at most 1/K times the minimum edge capacity for some K > 1. Here we provide combinatorial algorithms that match the currently best known algorithms. All of our algorithms are strongly polynomial and some can even be used in the online setting.",
keywords = "Combinatorial algorithms, Unsplittable flow problem",
author = "Yossi Azar and Oded Regev",
year = "2006",
month = "1",
doi = "10.1007/s00453-005-1172-z",
language = "English (US)",
volume = "44",
pages = "49--66",
journal = "Algorithmica",
issn = "0178-4617",
publisher = "Springer New York",
number = "1",

}

TY - JOUR

T1 - Combinatorial algorithms for the unsplittable flow problem

AU - Azar, Yossi

AU - Regev, Oded

PY - 2006/1

Y1 - 2006/1

N2 - We provide combinatorial algorithms for the unsplittable flow problem (UFP) that either match or improve the previously best results. In the UFP we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path subject to the capacity constraints such that the total profit of the connected pairs is maximized.We consider three variants of the problem. First is the classical UFP in which the maximum demand is at most the minimum edge capacity. It was previously known to have an O(√m) approximation algorithm; the algorithm is based on the randomized rounding technique and its analysis makes use of the Chernoff bound and the FKG inequality.We provide a combinatorial algorithm that achieves the same approximation ratio and whose analysis is considerably simpler. Second is the extended UFP in which some demands might be higher than edge capacities. Our algorithm for this case improves the best known approximation ratio. We also give a lower bound that shows that the extended UFP is provably harder than the classical UFP. Finally, we consider the bounded UFP in which the maximum demand is at most 1/K times the minimum edge capacity for some K > 1. Here we provide combinatorial algorithms that match the currently best known algorithms. All of our algorithms are strongly polynomial and some can even be used in the online setting.

AB - We provide combinatorial algorithms for the unsplittable flow problem (UFP) that either match or improve the previously best results. In the UFP we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path subject to the capacity constraints such that the total profit of the connected pairs is maximized.We consider three variants of the problem. First is the classical UFP in which the maximum demand is at most the minimum edge capacity. It was previously known to have an O(√m) approximation algorithm; the algorithm is based on the randomized rounding technique and its analysis makes use of the Chernoff bound and the FKG inequality.We provide a combinatorial algorithm that achieves the same approximation ratio and whose analysis is considerably simpler. Second is the extended UFP in which some demands might be higher than edge capacities. Our algorithm for this case improves the best known approximation ratio. We also give a lower bound that shows that the extended UFP is provably harder than the classical UFP. Finally, we consider the bounded UFP in which the maximum demand is at most 1/K times the minimum edge capacity for some K > 1. Here we provide combinatorial algorithms that match the currently best known algorithms. All of our algorithms are strongly polynomial and some can even be used in the online setting.

KW - Combinatorial algorithms

KW - Unsplittable flow problem

UR - http://www.scopus.com/inward/record.url?scp=29544452472&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=29544452472&partnerID=8YFLogxK

U2 - 10.1007/s00453-005-1172-z

DO - 10.1007/s00453-005-1172-z

M3 - Article

AN - SCOPUS:29544452472

VL - 44

SP - 49

EP - 66

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 1

ER -