License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2015.406
URN: urn:nbn:de:0030-drops-49304
URL: https://drops.dagstuhl.de/opus/volltexte/2015/4930/
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### Minimum Cost Flows in Graphs with Unit Capacities

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### Abstract

We consider the minimum cost flow problem on graphs with unit capacities and its special cases. In previous studies, special purpose algorithms exploiting the fact that capacities are one have been developed.
In contrast, for maximum flow with unit capacities, the best bounds are proven for slight modifications of classical blocking flow and push-relabel algorithms.

In this paper we show that the classical cost scaling algorithms of Goldberg and Tarjan (for general integer capacities) applied to a problem with unit capacities achieve or improve the best known bounds.
For weighted bipartite matching we establish a bound of O(\sqrt{rm}\log C) on a slight variation of this algorithm. Here r is the size of the smaller side of the bipartite graph, m is the number of edges, and C is the largest absolute value of an arc-cost. This simplifies a result of [Duan et al. 2011] and improves the bound, answering an open question of [Tarjan and Ramshaw 2012]. For graphs with unit vertex capacities we establish a novel O(\sqrt{n}m\log(nC)) bound. We also give the first cycle canceling algorithm for minimum cost flow with unit capacities. The algorithm naturally generalizes the single source shortest path algorithm of [Goldberg 1995].

### BibTeX - Entry

@InProceedings{goldberg_et_al:LIPIcs:2015:4930,
author =	{Andrew V. Goldberg and Haim Kaplan and Sagi Hed and Robert E. Tarjan},
title =	{{Minimum Cost Flows in Graphs with Unit Capacities}},
booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
pages =	{406--419},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-78-1},
ISSN =	{1868-8969},
year =	{2015},
volume =	{30},
editor =	{Ernst W. Mayr and Nicolas Ollinger},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/4930},
URN =		{urn:nbn:de:0030-drops-49304},
doi =		{10.4230/LIPIcs.STACS.2015.406},
annote =	{Keywords: minimum cost flow, bipartite matching, unit capacity, cost scaling}
}


 Keywords: minimum cost flow, bipartite matching, unit capacity, cost scaling Collection: 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015) Issue Date: 2015 Date of publication: 26.02.2015

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