 When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2016.28
URN: urn:nbn:de:0030-drops-67989
URL: https://drops.dagstuhl.de/opus/volltexte/2016/6798/
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### Bipartite Matching with Linear Edge Weights

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### Abstract

Consider a complete weighted bipartite graph G in which each left vertex u has two real numbers intercept and slope, each right vertex v has a real number quality, and the weight of any edge (u, v) is defined as the intercept of u plus the slope of u times the quality of v. Let m (resp., n) denote the number of left (resp., right) vertices, and assume that m geq n. We develop a fast algorithm for computing a maximum weight matching (MWM) of such a graph. Our algorithm begins by computing an MWM of the subgraph induced by the n right vertices and an arbitrary subset of n left vertices; this step is straightforward to perform in O(n log n) time. The remaining m - n left vertices are then inserted into the graph one at a time, in arbitrary order. As each left vertex is inserted, the MWM is updated. It is relatively straightforward to process each such insertion in O(n) time; our main technical contribution is to improve this time bound to O(sqrt{n} log^2 n). This result has an application related to unit-demand auctions. It is well known that the VCG mechanism yields a suitable solution (allocation and prices) for any unit-demand auction. The graph G may be viewed as encoding a special kind of unit-demand auction in which each left vertex u represents a unit-demand bid, each right vertex v represents an item, and the weight of an edge (u, v) represents the offer of bid u on item v. In this context, our fast insertion algorithm immediately provides an O(sqrt{n} log^2 n)-time algorithm for updating a VCG allocation when a new bid is received. We show how to generalize the insertion algorithm to update (an efficient representation of) the VCG prices within the same time bound.

### BibTeX - Entry

```@InProceedings{domanic_et_al:LIPIcs:2016:6798,
author =	{Nevzat Onur Domanic and Chi-Kit Lam and C. Gregory Plaxton},
title =	{{Bipartite Matching with Linear Edge Weights}},
booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
pages =	{28:1--28:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-026-2},
ISSN =	{1868-8969},
year =	{2016},
volume =	{64},
editor =	{Seok-Hee Hong},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
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