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# Bipartite Matching with Linear Edge Weights

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## Cite As

Nevzat Onur Domanic, Chi-Kit Lam, and C. Gregory Plaxton. Bipartite Matching with Linear Edge Weights. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ISAAC.2016.28

## 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

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## References

1. G. Demange, D. Gale, and M. A. O. Sotomayor. Multi-item auctions. The Journal of Political Economy, 94:863-872, 1986.
2. N. O. Domaniç, C.-K. Lam, and C. G. Plaxton. Bipartite matching with linear edge weights. Technical Report TR-16-15, Department of Computer Science, University of Texas at Austin, October 2016.
3. N. O. Domaniç and C. G. Plaxton. Scheduling unit jobs with a common deadline to minimize the sum of weighted completion times and rejection penalties. In Proceedings of the 25th International Symposium on Algorithms and Computation, pages 646-657, 2014.
4. R. Duan and H.-H. Su. A scaling algorithm for maximum weight matching in bipartite graphs. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1413-1424, 2012.
5. M. L. Fredman and R. E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM, 34:596-615, 1987.
6. H. N. Gabow and R. E. Tarjan. A linear-time algorithm for a special case of disjoint set union. Journal of Computer and System Sciences, 30:209-221, 1985.
7. F. Glover. Maximum matching in a convex bipartite graph. Naval Research Logistics Quarterly, 14:313-316, 1967.
8. J. Green and J.-J. Laffont. Characterization of satisfactory mechanisms for the revelation of preferences for public goods. Econometrica, 45:427-438, 1977.
9. G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities. Cambridge University Press, 2nd edition, 1952.
10. I. Katriel. Matchings in node-weighted convex bipartite graphs. INFORMS Journal on Computing, 20:205-211, 2008.
11. H. W. Kuhn. The Hungarian method for the assignment problem. Naval Research Logistics Quarterly, 2:83-97, 1955.
12. H. B. Leonard. Elicitation of honest preferences for the assignment of individuals to positions. The Journal of Political Economy, 91:461-479, 1983.
13. W. Lipski, Jr. and F. P. Preparata. Efficient algorithms for finding maximum matchings in convex bipartite graphs and related problems. Acta Informatica, 15:329-346, 1981.
14. C. G. Plaxton. Vertex-weighted matching in two-directional orthogonal ray graphs. In Proceedings of the 24th International Symposium on Algorithms and Computation, pages 524-534, 2013.
15. G. Steiner and J. S. Yeomans. A linear time algorithm for maximum matchings in convex, bipartite graphs. Computers and Mathematics with Applications, 31:91-96, 1996.
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