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A Tight Max-Flow Min-Cut Duality Theorem for Non-Linear Multicommodity Flows [article]

Matthew Broussard, Bala Krishnamoorthy
2021 arXiv   pre-print
The Max-Flow Min-Cut theorem is the classical duality result for the Max-Flow problem, which considers flow of a single commodity.  ...  Furthermore, we present a simple class of the multicommodity max flow problem where computations using this tight duality result could run significantly faster than default brute force computations.  ...  Acknowledgment We thank the National Science Foundation for support through grants 1661348 and 1819229.  ... 
arXiv:2107.04252v1 fatcat:yclotrhmjjfm7ioyl5r4aalhh4

Multiway cuts in node weighted graphs

Naveen Garg, Vijay V. Vazirani, Mihalis Yannakakis
2004 Journal of Algorithms  
A (2 -2/k) approximation algorithm is presented for the node multiway cut problem, thus matching the result of Dahlhaus et al. (SIAM J.  ...  Comput. 23 (4) (1994) 864-894) for the edge version of this problem. This is done by showing that the associated LP-relaxation always has a half-integral optimal solution.  ...  Approximate max-flow min-multiway cut theorem Theorem 5 (Approximate max-flow min-multiway cut theorem). ( 2 \ maximum flow ^ minimum node multiway cut ^ I 2 2 • maximum flow.  ... 
doi:10.1016/s0196-6774(03)00111-1 fatcat:jxfpih7nezhebbrj5q7p2bcxzq

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

Petr Kolman, Christian Scheideler
2013 Theory of Computing Systems  
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  ...  An elementary h-route flow, for an integer h ≥ 1, is a set of h edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an h-route flow is a non-negative linear combination  ...  The first author would like to thank Jiří Sgall and Thomas Erlebach for stimulating discussions.  ... 
doi:10.1007/s00224-013-9454-3 fatcat:vjlestmhavezhmuumfcb5bw6uy

An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm

Yonatan Aumann, Yuval Rabani
1998 SIAM journal on computing (Print)  
An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent flow, and thus of the optimal min cut ratio, is presented.  ...  It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for k-commodity flow instances with arbitrary capacities and demands.  ...  Min-Cut-Ratio Max-Concurrent-Flow Let G = (V, E) be a graph. Let c : E → IR + be an assignment of non-negative capacities to the edges of G.  ... 
doi:10.1137/s0097539794285983 fatcat:i6bjolss7bf4lf4tng53dmckma

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

Min-Max Results in Combinatorial Optimization [chapter]

A. Schrijver
1983 Mathematical Programming The State of the Art  
This relation implies Konig's and Gupta's edge-colouring theorems (Theorem 3 and Corollary 3 a). Acknowledgements. I thank Dr. W. Cook and Dr. W. R. Pulleyblank for their helpful comments.  ...  (20) min {k I there exist pairwise disjoint subsets Si. ... , Sk of S such that, for all i = 1, 2 and S' E 'lf;, S' intersects at least g; (S') of the S;)=max{g;(S')li= 1, 2; S'E'IB';} (assuming g;  ...  ., in the max-flow min-cut problem each of the arcs in the minimum-capacitated cut has a tight capacity.  ... 
doi:10.1007/978-3-642-68874-4_18 dblp:conf/ismp/Schrijver82 fatcat:jnz2kyrjnzfnlpwxbkondscuki

Solving nonlinear multicommodity flow problems by the analytic center cutting plane method

J. -L. Goffin, J. Gondzio, R. Sarkissian, J. -P. Vial
1997 Mathematical programming  
The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them.  ...  Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network.  ...  Acknowledgement The authors thank Olivier du Merle for many helpful suggestions concerning the efficient implementation of ACCPM and for running the Dantzig-Wolfe algorithm.  ... 
doi:10.1007/bf02614381 fatcat:dif2rr4uzjdmvfm26fdv6krwvq

Eigenvalue bounds, spectral partitioning, and metrical deformations via flows

Punyashloka Biswal, James R. Lee, Satish Rao
2010 Journal of the ACM  
Moreover, while the standard "sweep" algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary  ...  We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs.  ...  Acknowledgements We thank Oded Schramm for helpful pointers to the literature on discrete conformal mappings.  ... 
doi:10.1145/1706591.1706593 fatcat:6depqdz32vcvfjulj6udtfj2rq

Eigenvalue Bounds, Spectral Partitioning, and Metrical Deformations via Flows

Punyashloka Biswal, James R. Lee, Satish Rao
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
Moreover, while the standard "sweep" algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary  ...  We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs.  ...  Acknowledgements We thank Oded Schramm for helpful pointers to the literature on discrete conformal mappings.  ... 
doi:10.1109/focs.2008.78 dblp:conf/focs/BiswalLR08 fatcat:rdpqk3dnbrcldetyfmwxovyeai

Eigenvalue bounds, spectral partitioning, and metrical deformations via flows [article]

Punyashloka Biswal, James R. Lee, Satish Rao
2008 arXiv   pre-print
Moreover, while the standard "sweep" algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary  ...  We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs.  ...  Acknowledgements We thank Oded Schramm for helpful pointers to the literature on discrete conformal mappings.  ... 
arXiv:0808.0148v2 fatcat:26cjz5u4ana3hcozybgreztsea

Length-bounded cuts and flows

Georg Baier, Thomas Erlebach, Alexander Hall, Ekkehard Köhler, Petr Kolman, Ondřej Pangrác, Heiko Schilling, Martin Skutella
2010 ACM Transactions on Algorithms  
For a given number L, an L-length-bounded edge-cut (node-cut, resp.) in a graph G with source s and sink t is a set C of edges (nodes, resp.) such that no s-t-path of length at most L remains in the graph  ...  We show that the minimum length-bounded cut problem is N P-hard to approximate within a factor of 1.1377 for L ≥ 5 in the case of nodecuts and for L ≥ 4 in the case of edge-cuts.  ...  ACKNOWLEDGMENT The authors residing in the lowlands would like to thank the author residing in the highlands for his sedulous commitment and readiness to make sacrifices for the sake of this work.  ... 
doi:10.1145/1868237.1868241 fatcat:2rp6wva4ovfhtlhcf7cswqwoze

Length-Bounded Cuts and Flows [chapter]

Georg Baier, Thomas Erlebach, Alexander Hall, Ekkehard Köhler, Heiko Schilling, Martin Skutella
2006 Lecture Notes in Computer Science  
For a given number L, an L-length-bounded edge-cut (node-cut, resp.) in a graph G with source s and sink t is a set C of edges (nodes, resp.) such that no s-t-path of length at most L remains in the graph  ...  We show that the minimum length-bounded cut problem is N P-hard to approximate within a factor of 1.1377 for L ≥ 5 in the case of nodecuts and for L ≥ 4 in the case of edge-cuts.  ...  ACKNOWLEDGMENT The authors residing in the lowlands would like to thank the author residing in the highlands for his sedulous commitment and readiness to make sacrifices for the sake of this work.  ... 
doi:10.1007/11786986_59 fatcat:brgpjys3mfhedmxtbgsrlxon5y

Polynomial flow-cut gaps and hardness of directed cut problems

Julia Chuzhoy, Sanjeev Khanna
2007 Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07  
Starting with the celebrated max flow-min cut theorem of Ford and Fulkerson, flow-cut gaps have played a central role in combinatorial optimization.  ...  The natural linear programming relaxation for multicut corresponds, by LP-duality, to the well-studied maximum (fractional) multicommodity flow problem, while the natural LP-relaxation for sparsest cut  ...  Acknowledgements We would like to thank Irit Dinur, Piotr Indyk, and Ran Raz for helpful discussions.  ... 
doi:10.1145/1250790.1250817 dblp:conf/stoc/ChuzhoyK07 fatcat:3odp4wfusjbcplf4hoeoyaallu

Polynomial flow-cut gaps and hardness of directed cut problems

Julia Chuzhoy, Sanjeev Khanna
2009 Journal of the ACM  
Starting with the celebrated max flow-min cut theorem of Ford and Fulkerson, flow-cut gaps have played a central role in combinatorial optimization.  ...  The natural linear programming relaxation for multicut corresponds, by LP-duality, to the well-studied maximum (fractional) multicommodity flow problem, while the natural LP-relaxation for sparsest cut  ...  Acknowledgements We would like to thank Irit Dinur, Piotr Indyk, and Ran Raz for helpful discussions.  ... 
doi:10.1145/1502793.1502795 fatcat:dcdikx475nffzimi55ffpm45ru

The all-or-nothing flow problem in directed graphs with symmetric demand pairs

Chandra Chekuri, Alina Ene
2014 Mathematical programming  
A subset M of the pairs is routable if there is a feasible multicommodity flow in G such that, for each pair s i t i ∈ M , the amount of flow from s i to t i is at least one and the amount of flow from  ...  We study the approximability of the All-or-Nothing multicommodity flow problem in directed graphs with symmetric demand pairs (SymANF).  ...  A linear relationship would have applications to routing on disjoint paths, but would also give an improved upper bound on the flow-cut gap for symmetric product multicommodity flows in planar directed  ... 
doi:10.1007/s10107-014-0856-z fatcat:l4cjh42tf5evbeprkaikfqwlte
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