On complexity, representation and approximation of integral multicommodity flows

The paper has two parts. In the algorithmic part integer inequality systems of packing types and their application to integral multicommodity flow problems are considered. We give 1− ε approximation algorithms using the randomized rounding/derandomization scheme provided that the components of the r...

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Veröffentlicht in:Discrete Applied Mathematics Jg. 99; H. 1; S. 183 - 208
Hauptverfasser: Srivastav, Anand, Stangier, Peter
Format: Journal Article Tagungsbericht
Sprache:Englisch
Veröffentlicht: Lausanne Elsevier B.V 01.02.2000
Amsterdam Elsevier
New York, NY
Schlagworte:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Combinatorial probability
Computer science; control theory; systems
Derandomization
Distribution theory
Exact sciences and technology
Integer programming
Integral multicommodity flows
Mathematical programming
Mathematics
Operational research and scientific management
Operational research. Management science
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Theoretical computing
60E15
Integer programming
60C05
68Q25
90C35
Integral multicommodity flows
05C85
68R10
Derandomization
Randomization
Fractional flow
Multicomponent analysis
Network flow
Angluin Valiant inequality
Multicommodity flow
Algorithm
Computational complexity
ISSN:0166-218X, 1872-6771
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Abstract The paper has two parts. In the algorithmic part integer inequality systems of packing types and their application to integral multicommodity flow problems are considered. We give 1− ε approximation algorithms using the randomized rounding/derandomization scheme provided that the components of the right-hand side vector (resp. the capacities) are in Ω(ε −2 log m) where m is the number of constraints (resp. the number of edges). In the complexity-theoretic part it is shown that the approximable instances above build hard problems. Extending a result of Garg et al. (Algorithmica 18 (1997) 3–20), the non approximability of the maximum integral multicommodity flow problem for trees with a large capacity function c∈ Ω( log m) , is proved. Furthermore, for every fixed non-negative integer K the problem with specified demand function r− K is NP-hard even if c is any function polynomially bounded in n and if the problem with demand function r is fractionally solvable. For fractionally solvable multicommodity flow problems with nonplanar union of supply and demand graph the integrality gap is unbounded, while in the planar case Korach and Penn (Technical Report, Computer Science Department, Israel Institute of Technology, Haifa, 1989) could fix it to 1. Finally, an interesting relation between discrepancies of set systems and integral multicommodity flows with specified demands is discussed.
AbstractList The paper has two parts. In the algorithmic part integer inequality systems of packing types and their application to integral multicommodity flow problems are considered. We give 1− ε approximation algorithms using the randomized rounding/derandomization scheme provided that the components of the right-hand side vector (resp. the capacities) are in Ω(ε −2 log m) where m is the number of constraints (resp. the number of edges). In the complexity-theoretic part it is shown that the approximable instances above build hard problems. Extending a result of Garg et al. (Algorithmica 18 (1997) 3–20), the non approximability of the maximum integral multicommodity flow problem for trees with a large capacity function c∈ Ω( log m) , is proved. Furthermore, for every fixed non-negative integer K the problem with specified demand function r− K is NP-hard even if c is any function polynomially bounded in n and if the problem with demand function r is fractionally solvable. For fractionally solvable multicommodity flow problems with nonplanar union of supply and demand graph the integrality gap is unbounded, while in the planar case Korach and Penn (Technical Report, Computer Science Department, Israel Institute of Technology, Haifa, 1989) could fix it to 1. Finally, an interesting relation between discrepancies of set systems and integral multicommodity flows with specified demands is discussed.
Author Stangier, Peter
Srivastav, Anand
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  givenname: Peter
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  email: pstangier@mac.de
  organization: Zentrum für Paralelles Rechnen, Universität zu Köln, Weyertal 80, D-50931 Köln, Germany
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Cites_doi 10.1007/BF02523685
10.1145/103418.103425
10.1002/(SICI)1098-2418(199601)8:1<27::AID-RSA2>3.0.CO;2-T
10.1017/CBO9780511814075
10.1287/opre.34.2.250
10.1016/0022-0000(79)90045-X
10.1137/0205048
10.1007/BF01294465
10.1016/S0095-8956(81)80012-3
10.1016/0022-0000(88)90003-7
10.1007/978-3-642-97881-4
10.1007/BF02579324
10.1016/0166-218X(93)90153-F
10.1016/0020-0190(78)90016-9
10.1017/CBO9781107359949.008
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Issue 1
Keywords 60E15
Integer programming
60C05
68Q25
90C35
Integral multicommodity flows
05C85
68R10
Derandomization
Randomization
Fractional flow
Multicomponent analysis
Network flow
Angluin Valiant inequality
Multicommodity flow
Algorithm
Computational complexity
Language English
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Snippet The paper has two parts. In the algorithmic part integer inequality systems of packing types and their application to integral multicommodity flow problems are...
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SubjectTerms Algorithmics. Computability. Computer arithmetics
Applied sciences
Combinatorial probability
Computer science; control theory; systems
Derandomization
Distribution theory
Exact sciences and technology
Integer programming
Integral multicommodity flows
Mathematical programming
Mathematics
Operational research and scientific management
Operational research. Management science
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Theoretical computing
Title On complexity, representation and approximation of integral multicommodity flows
URI https://dx.doi.org/10.1016/S0166-218X(99)00133-X
Volume 99
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