NDLI: Edge-disjoint paths revisited

Content Provider	ACM Digital Library
Author	Khanna, Sanjeev Chekuri, Chandra
Copyright Year	2007
Abstract	The approximability of the maximum edge-disjoint paths problem (EDP) in directed graphs was seemingly settled by an $Ω(m^{1/2}$ ™ &epsis;)-hardness result of Guruswami et al. [2003], and an $\textit{O}(&sqrt;\textit{m})$ approximation achievable via a natural multicommodity-flow-based LP relaxation as well as a greedy algorithm. Here $\textit{m}$ is the number of edges in the graph. We observe that the $Ω(m^{1/2}$ ™ &epsis;)-hardness of approximation applies to $\textit{sparse}$ graphs, and hence when expressed as a function of $\textit{n},$ that is, the number of vertices, only an $Ω(n^{1/2}™$ &epsis;)-hardness follows. On the other hand, $\textit{O}(&sqrt;\textit{m})-approximation$ algorithms do not guarantee a $\textit{sublinear}$ (in terms of $\textit{n})$ approximation algorithm for $\textit{dense}$ graphs. We note that a similar gap exists in the known results on the integrality gap of the flow-based LP relaxation: an $Ω(&sqrt;\textit{n})$ lower bound and $\textit{O}(&sqrt;\textit{m})$ upper bound. Motivated by this discrepancy in the upper and lower bounds, we study algorithms for EDP in directed and undirected graphs and obtain improved approximation ratios. We show that the greedy algorithm has an approximation ratio of $O(min(n^{2/3},$ $&sqrt;\textit{m}))$ in undirected graphs and a ratio of $O(min(n^{4/5},$ $&sqrt;\textit{m}))$ in directed graphs. For acyclic graphs we give an $\textit{O}(&sqrt;\textit{n}$ ln $\textit{n})$ approximation via LP rounding. These are the first sublinear approximation ratios for EDP. The results also extend to EDP with weights and to the uniform-capacity unsplittable flow problem (UCUFP).
File Format	PDF
ISSN	15496325
e-ISSN	15496333
DOI	10.1145/1290672.1290683
Volume Number	3
Issue Number	4
Journal	ACM Transactions on Algorithms (TALG)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2007-11-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Edge-disjoint paths Approximation algorithm Greedy algorithm Multicommodity flow relaxation
Content Type	Text
Resource Type	Article
Subject	Mathematics

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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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