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### An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations [article]

Jonathan A. Kelner, Yin Tat Lee, Lorenzo Orecchia, Aaron Sidford
2013 arXiv   pre-print
In this paper, we introduce a new framework for approximately solving flow problems in capacitated, undirected graphs and apply it to provide asymptotically faster algorithms for the maximum s-t flow and  ...  We also note that independently Jonah Sherman produced an almost linear time algorithm for maximum flow and we thank him for coordinating submissions.  ...  Acknowledgements We thank Jonah Sherman for agreeing to coordinate submissions and we thank Satish Rao, Jonah Sherman, Daniel Spielman, Shang-Hua Teng.  ...

### An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations [chapter]

Jonathan A. Kelner, Yin Tat Lee, Lorenzo Orecchia, Aaron Sidford
2013 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
In this paper, we introduce a new framework for approximately solving flow problems in capacitated, undirected graphs and apply it to provide asymptotically faster algorithms for the maximum s-t flow and  ...  For graphs with n vertices and m edges, it allows us to find an ε-approximate maximum s-t flow in time O(m 1+o(1) ε −2 ), improving on the previous best bound of O(mn 1/3 poly(ε −1 )).  ...  Acknowledgements We thank Jonah Sherman for agreeing to coordinate arXiv postings, and we thank Satish Rao, Daniel Spielman, Shang-Hua Teng for many helpful conversations.  ...

### Approximation Algorithms for Multicommodity-Type Problems with Guarantees Independent of the Graph Size

Ankur Moitra
2009 2009 50th Annual IEEE Symposium on Foundations of Computer Science
ACKNOWLEDGMENTS We would like to thank Yossi Azar, Tom Leighton, Harald Räcke, Satish Rao and Mohit Singh for many helpful discussions.  ...  (in n and k) time O(log 4 k log log k)-approximation algorithm for the Steiner Minimum Linear Arrangement Problem, and a polynomial time O(log 4 k log log k)-approximation algorithm for the Steiner Minimum  ...  Theorem 9: There exist a polynomial (in n and k) time algorithm to construct an undirected, capacitated graph G = (K, E ) such that the cut function h of this graph O(log 3.5 k)-approximates the terminal  ...

### Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver

Zeyuan Allen-Zhu, Yin Tat Lee, Lorenzo Orecchia
2015 Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms
An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations. SODA'14: ACM-SIAM Proc. Symp. Discrete Algorithms, pp. 217-226, 2014.  ...  ACM. • Z.Allen-Zhu and L.Orecchia. Flow-Based Algorithms for Local Graph Clustering. SODA'14: ACM-SIAM Proc. Symp. Discrete Algorithms, pp. 1267-1286, 2014. Discrete Algorithms, pp. 1824-1831, 2016.  ...  Organizer of semester-long program "Bridging Continuous and Discrete Optimization" at the Simons Institute for Theoretical Computer Science, to run in Fall 2017.  ...

### Vertex Sparsification and Oblivious Reductions

Ankur Moitra
2013 SIAM journal on computing (Print)
These results allow us to derive a more general theory of Steiner cut and flow problems, and allow us to obtain approximation algorithms with guarantees independent of the size of the graph for a number  ...  Given an undirected, capacitated graph G = (V, E) and a set K ⊂ V of terminals of size k, we construct an undirected, capacitated graph G = (K, E ) for which the cut-function approximates the value of  ...  We would like to thank Tom Leighton, Harald Räcke and Satish Rao for many helpful discussions.  ...

### Algorithm Finding Maximum Concurrent Multicommodity Linear Flow with Limited Cost in Extended Traffic Network with Single Regulating Coefficient on Two-Side Lines

Tran Quoc Chien, Nguyen Dinh Lau
2017 International Journal of Computer Networks & Communications
Algorithms to find Maximum Concurrent Multicommodity Flow with Limited Cost on extended traffic networks are introduced in the works we did.  ...  This work studies the more general and practical case, where flows are limited to use two-sided lines with a single parameter called regulating coefficient.  ...  Algorithms to find approximate maximum concurrent multicommodity linear flow with limited cost are developed in the works [4, 5] .  ...

### Faster approximate multicommodity flow using quadratically coupled flows

Jonathan A. Kelner, Gary L. Miller, Richard Peng
2012 Proceedings of the 44th symposium on Theory of Computing - STOC '12
Given an undirected graph with m edges and k commodities, we give algorithms that find 1 − ǫ approximate solutions to the maximum concurrent flow problem and the maximum weighted multicommodity flow problem  ...  In this paper we present an adaptation of recent advances in single-commodity flow algorithms to this problem.  ...  A stronger result would be an approximation of minimum cost flow that runs inÕ(m 4/3 poly(k, ǫ −1 ) time. This problem is significantly harder due to it incorporating both L ∞ and L 1 constraints.  ...

### Greedy Edge-Disjoint Paths in Complete Graphs [chapter]

Paz Carmi, Thomas Erlebach, Yoshio Okamoto
2003 Lecture Notes in Computer Science
In fact, Kolman{Scheideler 9] gave the \shortening lemma" for the unsplittable ow problem (a generalization of the maximum edge-disjoint path problem) and it was used for improving the approximation ratio  ...  We consider MEDP in complete graphs. Erlebach{Vukadinovi c 4] showed that MEDP in complete graphs is NPhard and has a constant-factor approximation algorithm.  ...  Acknowledgement The author is deeply grateful to Thomas Erlebach for his advice and kindness.  ...

### Approximating Directed Multicuts

Joseph Cheriyan*, Howard Karloff, Yuval Rabani†
2005 Combinatorica
method for designing approximation algorithms, and generated novel tools for utilizing linear programming relaxations.  ...  The seminal paper of Leighton and Rao (1988) and subsequent papers presented approximate minmax theorems relating multicommodity flow values and cut capacities in undirected networks, developed the divide-and-conquer  ...  We thank Joel Friedman and Mihalis Yannakakis for their help in the early phase of our work and we thank Aravind Srinivasan for his comments.  ...

### Graph partitioning using single commodity flows

Rohit Khandekar, Satish Rao, Umesh Vazirani
2009 Journal of the ACM
We show that the sparsest cut in graphs can be approximated within Ç´ÐÓ ¾ Òµ factor in Ç´Ò ¿ ¾ µ time using polylogarithmic single commodity max-flow computations.  ...  Previous algorithms are based on multicommodity flows which take time Ç´Ò ¾ µ.  ...  Thus it also computes an Ç´ÐÓ ¾ Òµ approximation to the uniform concurrent multicommodity flows in time Ç´Ñ · Ò ¿ ¾ µ.  ...

### Graph partitioning using single commodity flows

Rohit Khandekar, Satish Rao, Umesh Vazirani
2006 Proceedings of the thirty-eighth annual ACM symposium on Theory of computing - STOC '06
We show that the sparsest cut in graphs can be approximated within Ç´ÐÓ ¾ Òµ factor in Ç´Ò ¿ ¾ µ time using polylogarithmic single commodity max-flow computations.  ...  Previous algorithms are based on multicommodity flows which take time Ç´Ò ¾ µ.  ...  Thus it also computes an Ç´ÐÓ ¾ Òµ approximation to the uniform concurrent multicommodity flows in time Ç´Ñ · Ò ¿ ¾ µ.  ...

### Faster Approximate Multicommodity Flow Using Quadratically Coupled Flows [article]

Jonathan A. Kelner, Gary Miller, Richard Peng
2012 arXiv   pre-print
Given an undirected graph with m edges and k commodities, we give algorithms that find 1-ϵ approximate solutions to the maximum concurrent flow problem and the maximum weighted multicommodity flow problem  ...  In this paper we present an adaptation of recent advances in single-commodity flow algorithms to this problem.  ...  This allows us to approximate multicommodity flows in timeÕ(m 4/3 poly(k, ǫ −1 )).  ...

### Maximum k-Splittable s,t-Flows

Ronald Koch, Martin Skutella, Ines Spenke
2007 Theory of Computing Systems
If k is part of the input, this problem is still N P -hard and we present a polynomial time approximation scheme for it.  ...  We give efficient algorithms for computing exact and approximate solutions by decoupling the two decisions into a first packing step and a second routing step.  ...  Acknowledgement We thank two anonymous referees for many useful comments that helped to improve the presentation of the paper.  ...

### A polynomial-time tree decomposition to minimize congestion

Chris Harrelson, Kirsten Hildrum, Satish Rao
2003 Proceedings of the fifteenth annual ACM symposium on Parallel algorithms and architectures - SPAA '03
Räcke recently gave a remarkable proof showing that any undirected multicommodity flow problem can be routed in an oblivious fashion with congestion that is within a factor of O(log 3 n) of the best off-line  ...  Their construction was not based on finding a hierarchical decomposition, and this precludes its application to iterative methods for solving linear systems.  ...  Their result applied to directed graphs as well as undirected graphs, and they relied on Räcke's result to give an upper bound on the relationship between the best offline and oblivious solutions for undirected  ...

### A polynomial-time tree decomposition to minimize congestion

Chris Harrelson, Kirsten Hildrum, Satish Rao
2003 Proceedings of the fifteenth annual ACM symposium on Parallel algorithms and architectures - SPAA '03
Räcke recently gave a remarkable proof showing that any undirected multicommodity flow problem can be routed in an oblivious fashion with congestion that is within a factor of O(log 3 n) of the best off-line  ...  Their construction was not based on finding a hierarchical decomposition, and this precludes its application to iterative methods for solving linear systems.  ...  Their result applied to directed graphs as well as undirected graphs, and they relied on Räcke's result to give an upper bound on the relationship between the best offline and oblivious solutions for undirected  ...
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