Computing Maximum Flow with Augmenting Electrical Flows
We present an Õ (m 7/10 U 1/7)-time algorithm for the maximum s-t flow problem (and the minimum s-t cut problem) in directed graphs with m arcs and largest integer capacity U. This matches the running time of the Õ (mU)10/7)- time algorithm of Madry [30] in the unit-capacity case, and improves ove...
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| Format: | Article |
| Language: | en_US |
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Institute of Electrical and Electronics Engineers (IEEE)
2018
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| Online Access: | http://hdl.handle.net/1721.1/113573 https://orcid.org/0000-0003-0536-0323 |
| _version_ | 1826215870396891136 |
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| author | Madry, Aleksander |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Madry, Aleksander |
| author_sort | Madry, Aleksander |
| collection | MIT |
| description | We present an Õ (m 7/10 U 1/7)-time algorithm for the maximum s-t flow problem (and the minimum s-t cut problem) in directed graphs with m arcs and largest integer capacity U. This matches the running time of the Õ (mU)10/7)- time algorithm of Madry [30] in the unit-capacity case, and improves over it, as well as over the Õ (m√n log U)-time algorithm of Lee and Sidford [25], whenever U is moderately large and the graph is sufficiently sparse. By well-known reductions, this also implies similar running time improvements for the maximum-cardinality bipartite b-matching problem. One of the advantages of our algorithm is that it is significantly simpler than the ones presented in [30] and [25]. In particular, these algorithms employ a sophisticated interior-point method framework, while our algorithm is cast directly in the classic augmenting path setting that almost all the combinatorial maximum flow algorithms use. At a high level, the presented algorithm takes a primal dual approach in which each iteration uses electrical flows computations both to find an augmenting s-t flow in the current residual graph and to update the dual solution. We show that by maintain certain careful coupling of these primal and dual solutions we are always guaranteed to make significant progress. |
| first_indexed | 2024-09-23T16:38:21Z |
| format | Article |
| id | mit-1721.1/113573 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:38:21Z |
| publishDate | 2018 |
| publisher | Institute of Electrical and Electronics Engineers (IEEE) |
| record_format | dspace |
| spelling | mit-1721.1/1135732022-10-03T07:18:29Z Computing Maximum Flow with Augmenting Electrical Flows Madry, Aleksander Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Madry, Aleksander We present an Õ (m 7/10 U 1/7)-time algorithm for the maximum s-t flow problem (and the minimum s-t cut problem) in directed graphs with m arcs and largest integer capacity U. This matches the running time of the Õ (mU)10/7)- time algorithm of Madry [30] in the unit-capacity case, and improves over it, as well as over the Õ (m√n log U)-time algorithm of Lee and Sidford [25], whenever U is moderately large and the graph is sufficiently sparse. By well-known reductions, this also implies similar running time improvements for the maximum-cardinality bipartite b-matching problem. One of the advantages of our algorithm is that it is significantly simpler than the ones presented in [30] and [25]. In particular, these algorithms employ a sophisticated interior-point method framework, while our algorithm is cast directly in the classic augmenting path setting that almost all the combinatorial maximum flow algorithms use. At a high level, the presented algorithm takes a primal dual approach in which each iteration uses electrical flows computations both to find an augmenting s-t flow in the current residual graph and to update the dual solution. We show that by maintain certain careful coupling of these primal and dual solutions we are always guaranteed to make significant progress. 2018-02-12T16:03:59Z 2018-02-12T16:03:59Z 2016-12 2016-10 Article http://purl.org/eprint/type/ConferencePaper 978-1-5090-3933-3 http://hdl.handle.net/1721.1/113573 Madry, Aleksander. "Computing Maximum Flow with Augmenting Electrical Flows." 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), 9-11 October, 2016, New Brunswick, New Jersery, IEEE, 2016, pp. 593–602. https://orcid.org/0000-0003-0536-0323 en_US http://dx.doi.org/10.1109/FOCS.2016.70 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS) Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Institute of Electrical and Electronics Engineers (IEEE) MIT Web Domain |
| spellingShingle | Madry, Aleksander Computing Maximum Flow with Augmenting Electrical Flows |
| title | Computing Maximum Flow with Augmenting Electrical Flows |
| title_full | Computing Maximum Flow with Augmenting Electrical Flows |
| title_fullStr | Computing Maximum Flow with Augmenting Electrical Flows |
| title_full_unstemmed | Computing Maximum Flow with Augmenting Electrical Flows |
| title_short | Computing Maximum Flow with Augmenting Electrical Flows |
| title_sort | computing maximum flow with augmenting electrical flows |
| url | http://hdl.handle.net/1721.1/113573 https://orcid.org/0000-0003-0536-0323 |
| work_keys_str_mv | AT madryaleksander computingmaximumflowwithaugmentingelectricalflows |