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A Tight Max-Flow Min-Cut Duality Theorem for Non-Linear Multicommodity Flows
[article]
2021
arXiv
pre-print
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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
2004
Journal of Algorithms
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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
2013
Theory of Computing Systems
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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
1998
SIAM journal on computing (Print)
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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
1999
Journal of the ACM
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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]
1983
Mathematical Programming The State of the Art
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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
1997
Mathematical programming
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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
2010
Journal of the ACM
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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
2008
2008 49th Annual IEEE Symposium on Foundations of Computer Science
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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]
2008
arXiv
pre-print
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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
2010
ACM Transactions on Algorithms
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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]
2006
Lecture Notes in Computer Science
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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
2007
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07
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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
2009
Journal of the ACM
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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
2014
Mathematical programming
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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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