Negative-Weight shortest paths and unit capacity minimum cost flow in Õ(m[superscript 10/7] log W) Time
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Madry_Negative-weight.pdf
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Author(s)
Sankowski, Piotr
Cohen, Michael B.
Madry, Aleksander
Vladu, Adrian Valentin
Date Issued
January 2017
Journal
Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithm
Publisher
Association for Computing Machinery
Citation
Cohen, Michael B. et al. "Negative-weight shortest paths and unit capacity minimum cost flow in Õ(m[superscript 10/7] log W) time." Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithm, 16-19 January 2017, Barcelona, Spain, Association for Computing Machinery, 2017. pp. 752-771.
Version
Original manuscript
Abstract
In this paper, we study a set of combinatorial optimization problems on weighted graphs: the shortest path problem with negative weights, the weighted perfect bipartite matching problem, the unit-capacity minimum-cost maximum flow problem, and the weighted perfect bipartite b-matching problem under the assumption that ||b||1 = O(m). We show that each of these four problems can be solved in Õ(m[superscript 10/7] log W) time, where W is the absolute maximum weight of an edge in the graph, providing the first polynomial improvement in their sparse-graph time complexity in over 25 years.
At a high level, our algorithms build on the interior-point method-based framework developed by Mądry (FOCS 2013) for solving unit-capacity maximum flow problem. We develop a refined way to analyze this framework, as well as provide new variants of the underlying preconditioning and perturbation techniques. Consequently, we are able to extend the whole interior-point method-based approach to make it applicable in the weighted graph regime.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
http://dl.acm.org/citation.cfm?id=3039734
