A polynomial time approximation algorithm for the two-commodity splittable flow problem

We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity can be split into a bounded number k i of equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP...

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Vydáno v:	Mathematical methods of operations research (Heidelberg, Germany) Ročník 77; číslo 3; s. 381 - 391
Hlavní autoři:	Eisenschmidt, Elke, Haus, Utz-Uwe
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2013 Springer Springer Nature B.V
Témata:	Algorithms Approximation Business and Management Calculus of Variations and Optimal Control; Optimization Commodities Graphs Marketing Mathematical analysis Mathematics Mathematics and Statistics Operations research Operations Research/Decision Theory Original Article Restrictions Studies Surface hardness Splittable flow Approximation algorithm 2-commodity flow
ISSN:	1432-2994, 1432-5217
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Abstract	We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity can be split into a bounded number k i of equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP-hard, and hard to approximate to within a factor of α > 1/2. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even k i and a mild cut condition it can be modified to yield an exact method. The uniform case can be used to derive a 1/4-approximation for the maximum concurrent ( k 1 , k 2 )-splittable flow without chunk size restrictions for fixed demand ratios.
AbstractList	We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity $${i \in \{1, 2\}}$$ can be split into a bounded number k sub( )iof equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP-hard, and hard to approximate to within a factor of alpha > 1/2. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even k sub( )iand a mild cut condition it can be modified to yield an exact method. The uniform case can be used to derive a 1/4-approximation for the maximum concurrent (k sub(1), k sub(2))-splittable flow without chunk size restrictions for fixed demand ratios. We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity ... can be split into a bounded number k ^sub i^ of equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP-hard, and hard to approximate to within a factor of [alpha] > 1/2. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even k ^sub i^ and a mild cut condition it can be modified to yield an exact method. The uniform case can be used to derive a 1/4-approximation for the maximum concurrent (k ^sub 1^, k ^sub 2^)-splittable flow without chunk size restrictions for fixed demand ratios.[PUBLICATION ABSTRACT] (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity can be split into a bounded number k i of equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP-hard, and hard to approximate to within a factor of α > 1/2. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even k i and a mild cut condition it can be modified to yield an exact method. The uniform case can be used to derive a 1/4-approximation for the maximum concurrent ( k 1 , k 2 )-splittable flow without chunk size restrictions for fixed demand ratios.
Author	Eisenschmidt, Elke Haus, Utz-Uwe
Author_xml	– sequence: 1 givenname: Elke surname: Eisenschmidt fullname: Eisenschmidt, Elke organization: Institut für Mathematische Optimierung, Otto-von-Guericke Universität Magdeburg – sequence: 2 givenname: Utz-Uwe surname: Haus fullname: Haus, Utz-Uwe email: uhaus@math.ethz.ch organization: Institut für Operations Research, ETH Zürich
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Keywords	Splittable flow Approximation algorithm 2-commodity flow
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References	Baier G, Köhler E, Skutella M (2002) On the k-splittable flow problem. In: Algorithms—ESA 2002. Lecture notes in computer science, vol. 2461. Springer, Berlin, pp 101–113. doi:10.1007/3-540-45749-6_13 EvenSItaiAShamirAOn the complexity of timetable and multicommodity flow problemsSIAM J Comput1976546917034719740358.9002110.1137/0205048 Kleinberg JM (1996) Approximation algorithms for disjoint paths problems. Ph.d. thesis, Massachusetts Institute of Technology HuTCMulti-commodity network flowsOper Res19631133443600123.2370410.1287/opre.11.3.344 Schrijver A (2003) Combinatorial optimization. Polyhedra and efficiency. Vol. C, Algorithms and Combinatorics, vol. 24. Springer, Berlin BaierGKöhlerESkutellaMThe k-splittable flow problemAlgorithmica20054223124821469991086.9000710.1007/s00453-005-1167-9 402_CR1 TC Hu (402_CR4) 1963; 11 S Even (402_CR3) 1976; 5 G Baier (402_CR2) 2005; 42 402_CR5 402_CR6
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SubjectTerms	Algorithms Approximation Business and Management Calculus of Variations and Optimal Control; Optimization Commodities Graphs Marketing Mathematical analysis Mathematics Mathematics and Statistics Operations research Operations Research/Decision Theory Original Article Restrictions Studies Surface hardness
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Title	A polynomial time approximation algorithm for the two-commodity splittable flow problem
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