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 unitdemand auctions. It is well known that the VCG mechanism yields a suitable solution (allocation and prices) for any unitdemand auction. The graph G may be viewed as encoding a special kind of unitdemand auction in which each left vertex u represents a unitdemand 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 ChiKit Lam and C. Gregory Plaxton},
title = {{Bipartite Matching with Linear Edge Weights}},
booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)},
pages = {28:128:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770262},
ISSN = {18688969},
year = {2016},
volume = {64},
editor = {SeokHee Hong},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6798},
URN = {urn:nbn:de:0030drops67989},
doi = {10.4230/LIPIcs.ISAAC.2016.28},
annote = {Keywords: Weighted bipartite matching, Unitdemand auctions, VCG allocation and pricing}
}
Keywords: 

Weighted bipartite matching, Unitdemand auctions, VCG allocation and pricing 
Collection: 

27th International Symposium on Algorithms and Computation (ISAAC 2016) 
Issue Date: 

2016 
Date of publication: 

07.12.2016 