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Short proofs on multicommodity flows and cuts

A Schrijver
1991 Journal of combinatorial theory. Series B (Print)  
However, this contradicts the fact that Iw-)I = M-w I As is well known, this corollary implies a half-integral multicommodity flow theorem of Lomonosov [7] (extending the max-flow min-cut theorem of Ford  ...  As was noted by Karzanov [4] and Seymour 1143, cut packing results (like Karzanov's theorem above) can be interpreted in terms of polyhedral cones, and thus by polarity of cones are related to multicommodity  ... 
doi:10.1016/0095-8956(91)90052-l fatcat:biiskzq6affwrbvt3sbipwnlrm

Hallucination Helps: Energy Efficient Virtual Circuit Routing [chapter]

Antonios Antoniadis, Sungjin Im, Ravishankar Krishnaswamy, Benjamin Moseley, Viswanath Nagarajan, Kirk Pruhs, Cliff Stein
2013 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms  
The analysis of the online algorithm introduces a natural "priority" multicommodity flow problem, and bounds the priority multicommodity flow-cut gapthis might also be of independent interest.  ...  The analysis of the approximation ratio is then a direct consequence of the flow-cut gap for multicommodity flow.  ...  between concurrent multicommodity flow and sparsest cut [25, 26] ).  ... 
doi:10.1137/1.9781611973402.84 dblp:conf/soda/AntoniadisIKMNPS14 fatcat:ex2aqsgkjvgk3jdcamit4usor4

Edge-Disjoint Paths in Planar Graphs with Constant Congestion

Chandra Chekuri, Sanjeev Khanna, F. Bruce Shepherd
2009 SIAM journal on computing (Print)  
The natural multicommodity flow relaxation has an Ω( √ n) integrality gap where n is the number of nodes in G.  ...  Another ingredient we develop is a constant factor approximation for the all-or-nothing flow problem on OS instances via the flow relaxation.  ...  Chandra Chekuri and F. Bruce Shepherd acknowledge support from an ONR basic research grant N00014-05-1-0256 to Bell Labs.  ... 
doi:10.1137/060674442 fatcat:dcamyxlpyfanro6hkpisxtus7i

Edge-disjoint paths in Planar graphs with constant congestion

Chandra Chekuri, Sanjeev Khanna, F. Bruce Shepherd
2006 Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06  
The natural multicommodity flow relaxation has an Ω( √ n) integrality gap where n is the number of nodes in G.  ...  Another ingredient we develop is a constant factor approximation for the all-or-nothing flow problem on OS instances via the flow relaxation.  ...  Chandra Chekuri and F. Bruce Shepherd acknowledge support from an ONR basic research grant N00014-05-1-0256 to Bell Labs.  ... 
doi:10.1145/1132516.1132621 dblp:conf/stoc/ChekuriKS06 fatcat:o7cdx42fdzhxnnq7vnh5hfucma

A New Algorithm for Multicommodity Flow [article]

Dhananjay P. Mehendale
2010 arXiv   pre-print
We propose a new algorithm to obtain max flow for the multicommodity flow. This algorithm utilizes the max-flow min-cut theorem and the well known labeling algorithm due to Ford and Fulkerson [1].  ...  A record is made of these cuts and the paths flowing through the edges of these cuts. This record is then utilized to develop our algorithm to obtain max flow for multicommodity flow.  ...  Proof: The algorithm begins with applying max flow min cut theorem individually and separately to each source/sink pair of the given multicommodity flow and produces separate record of paths, min cuts,  ... 
arXiv:1001.0629v2 fatcat:yzkzashikfgm5egg66akw4heii

Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

Tom Leighton, Satish Rao
1999 Journal of the ACM  
In this paper, we establish max-flow min-cut theorems for several important classes of multicommodity flow problems.  ...  In particular, we show that for any n-node multicommodity flow problem with uniform demands, the max-flow for the problem is within an O(log n) factor of the upper bound implied by the min-cut.  ...  ACKNOWLEDGMENTS The proof of Theorem 7 was suggested by Eva Tardos. The fact that Theorem 2 could be extended in this way was also pointed out by David Shmoys and Alistair Sinclair.  ... 
doi:10.1145/331524.331526 fatcat:ec6h6h56brgn3hdcln3kkzoshy

Short length menger's theorem and reliable optical routing

Amitabha Bagchi, Amitabh Chaudhary, Petr Kolman
2003 Proceedings of the fifteenth annual ACM symposium on Parallel algorithms and architectures - SPAA '03  
In fact, this is only a corollary of a stronger result for multicommodity flow on networks with unit edge capacities: any multicommodity flow with k units for each commodity can be rerouted such that the  ...  We propose an O(k 2 F ) = O(k 2 −1 log n) approximation algorithm for this problem where F is the flow number of the graph, is the maximum degree and is the expansion.  ...  Acknowledgements We would like to thank Ankur Bhargava and Christian Scheideler for many useful discussions, and the anonymous referees for their helpful suggestions.  ... 
doi:10.1145/777449.777453 fatcat:kk2e7g7elbfjvfydl2xovqeu7q

Short length Menger's theorem and reliable optical routing

Amitabha Bagchi, Amitabh Chaudhary, Petr Kolman
2005 Theoretical Computer Science  
In fact, this is only a corollary of a stronger result for multicommodity flow on networks with unit edge capacities: any multicommodity flow with k units for each commodity can be rerouted such that the  ...  We propose an O(k 2 F ) = O(k 2 −1 log n) approximation algorithm for this problem where F is the flow number of the graph, is the maximum degree and is the expansion.  ...  Acknowledgements We would like to thank Ankur Bhargava and Christian Scheideler for many useful discussions, and the anonymous referees for their helpful suggestions.  ... 
doi:10.1016/j.tcs.2005.03.009 fatcat:wyd6f6qadjh4bic2isge7oaxde

Short length menger's theorem and reliable optical routing

Amitabha Bagchi, Amitabh Chaudhary, Petr Kolman
2003 Proceedings of the fifteenth annual ACM symposium on Parallel algorithms and architectures - SPAA '03  
In fact, this is only a corollary of a stronger result for multicommodity flow on networks with unit edge capacities: any multicommodity flow with k units for each commodity can be rerouted such that the  ...  We propose an O(k 2 F ) = O(k 2 −1 log n) approximation algorithm for this problem where F is the flow number of the graph, is the maximum degree and is the expansion.  ...  Acknowledgements We would like to thank Ankur Bhargava and Christian Scheideler for many useful discussions, and the anonymous referees for their helpful suggestions.  ... 
doi:10.1145/777412.777453 dblp:conf/spaa/BagchiCK03 fatcat:cooermxbmvhlrhhkw626prbx7q

An approximate max-flow min-cut relation for undirected multicommodity flow, with applications

Philip Klein, Satish Rao, Ajit Agrawal, R. Ravi
1995 Combinatorica  
In this paper, we prove the first approximate max-flow min-cut theorem for undirected multicommodity flow.  ...  and D is the sum of demands.  ...  We gratefully acknowledge helpful conversations with Tom Leighton, John Reif, David Shmoys, and ]~va Tardos. AN APPROXIMATE MAX-FLOW MIN-CUT THEOREM  ... 
doi:10.1007/bf01200755 fatcat:sdvzwb7w4nh47ioreboijkhg74

Multicommodity Flows in Planar Graphs with Demands on Faces [article]

Nikhil Kumar
2020 arXiv   pre-print
We consider the problem of multicommodity flows in planar graphs. Seymour showed that if the union of supply and demand graphs is planar, then the cut condition is sufficient for routing demands.  ...  We show that if the source sink pairs on each face of the graph are such that sources and sinks appear contiguously on the cycle bounding the face, then the flow cut gap is at most 3.  ...  Theorem 4 Let (G, H) be a separable face instance of multicommodity flow . Then, the flow-cut gap of (G, H) is at most 3. Proof. Follows from Lemma 2 and Theorem 3. Proof.  ... 
arXiv:2007.01280v1 fatcat:gm4nr6njajbilpcobn56rpxz34

Extensions and limits to vertex sparsification

F. Thomson Leighton, Ankur Moitra
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
Indirectly our result also allows us to give a construction for better quality cut sparsifiers (and flow sparsifiers).  ...  bounds against many multicommodity flows at once.  ...  on only short paths, using local paths.  ... 
doi:10.1145/1806689.1806698 dblp:conf/stoc/LeightonM10 fatcat:oniwcypynfdhhnmsegsug5fbqm

Page 572 of Mathematical Reviews Vol. 31, Issue 3 [page]

1966 Mathematical Reviews  
The main result is a generalization of the max-flow, min-cut theorem for multicommodity flows in bi-path networks.  ...  Bi-path networks and multicommodity flows. IEEE Trans. Cirewit Theory CT-11 (1964), 468-474.  ... 

Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing

Petr Kolman, Christian Scheideler
2013 Theory of Computing Systems  
Though the proof of the duality relies on this algorithm, it is not a straightforward corollary of it as in the case of classical multicommodity flows and cuts.  ...  The main result of this paper is an approximate duality theorem for multicommodity h-route cuts and flows, for h ≤ 3: The size of a minimum h-route cut is at least f /h and at most O(log 3 k ·f ) where  ...  The first author would like to thank Jiří Sgall and Thomas Erlebach for stimulating discussions.  ... 
doi:10.1007/s00224-013-9454-3 fatcat:vjlestmhavezhmuumfcb5bw6uy

Improving the multicommodity flow rates with network codes for two sources

Elona Erez, Meir Feder
2009 IEEE Journal on Selected Areas in Communications  
The scheme we suggest is based on modifying the multicommodity flow solution and thus improving the achievable rate region, w.r.t the uncoded case.  ...  For both the nondistributed case and the distributed case, the computational complexity of our algorithm for network coding is comparable to that of the parallel multicommodity flow problem.  ...  Improving the Multicommodity Flow Suppose we are given a rate pair (R 1 , R 2 ), which is on the boundary of the rate region of the multicommodity flow.  ... 
doi:10.1109/jsac.2009.090620 fatcat:izgyodngkfcrzaviykuprpe7na
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