{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9204","name":"Bipartite Matching with Linear Edge Weights","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.","keywords":["Weighted bipartite matching","Unit-demand auctions","VCG allocation and pricing"],"author":[{"@type":"Person","name":"Domanic, Nevzat Onur","givenName":"Nevzat Onur","familyName":"Domanic"},{"@type":"Person","name":"Lam, Chi-Kit","givenName":"Chi-Kit","familyName":"Lam"},{"@type":"Person","name":"Plaxton, C. Gregory","givenName":"C. Gregory","familyName":"Plaxton"}],"position":28,"pageStart":"28:1","pageEnd":"28:13","dateCreated":"2016-12-07","datePublished":"2016-12-07","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Domanic, Nevzat Onur","givenName":"Nevzat Onur","familyName":"Domanic"},{"@type":"Person","name":"Lam, Chi-Kit","givenName":"Chi-Kit","familyName":"Lam"},{"@type":"Person","name":"Plaxton, C. Gregory","givenName":"C. Gregory","familyName":"Plaxton"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ISAAC.2016.28","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6267","volumeNumber":64,"name":"27th International Symposium on Algorithms and Computation (ISAAC 2016)","dateCreated":"2016-12-07","datePublished":"2016-12-07","editor":{"@type":"Person","name":"Hong, Seok-Hee","givenName":"Seok-Hee","familyName":"Hong"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9204","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6267"}}}