Abstract
In this paper, we consider the online vertex-weighted bipartite matching problem in the random arrival model. We consider the generalization of the RANKING algorithm for this problem introduced by Huang, Tang, Wu, and Zhang [9], who show that their algorithm has a competitive ratio of 0.6534. We show that assumptions in their analysis can be weakened, allowing us to replace their derivation of a crucial function g on the unit square with a linear program that computes the values of a best possible g under these assumptions on a discretized unit square. We show that the discretization does not incur much error, and show computationally that we can obtain a competitive ratio of 0.6629. To compute the bound over our discretized unit square we use parallelization, and still needed two days of computing on a 64-core machine. Furthermore, by modifying our linear program somewhat, we can show computationally an upper bound on our approach of 0.6688; any further progress beyond this bound will require either further weakening in the assumptions of g or a stronger analysis than that of Huang et al.
B. Jin—Supported in part by NSERC fellowship PGSD3-532673-2019.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The full version of the paper can be accessed at https://arxiv.org/abs/2007.12823.
- 2.
We use notation for partial derivatives, but the result also holds for non-differentiable functions, if we use subgradients, etc. In particular,the result holds for the piecewise-affine functions g we obtain from solving the LP in Sect. 4. To keep the exposition simple, we will continue using partial derivative notation throughout the paper.
- 3.
We performed this computation on Amazon EC2. We used a compute-optimized c6g.16xlarge instance, running the Amazon Linux 2 AMI.
References
Aggarwal, G., Goel, G., Karande, C., Mehta, A.: Online vertex-weighted bipartite matching and single-bid budgeted allocations. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, pp. 1253–1264. Society for Industrial and Applied Mathematics (2011)
Bahmani, B., Kapralov, M.: Improved bounds for online stochastic matching. In: de Berg, M., Meyer, U. (eds.) ESA 2010. LNCS, vol. 6346, pp. 170–181. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15775-2_15
Birnbaum, B., Mathieu, C.: On-line bipartite matching made simple. SIGACT News 39, 81–87 (2008)
Brubach, B., Sankararaman, K.A., Srinivasan, A., Xu, P.: New algorithms, better bounds, and a novel model for online stochastic matching. CoRR, abs/1606.06395 (2016). http://arxiv.org/abs/1606.06395, arXiv:1606.06395
Devanur, N.R., Jain, K., Kleinberg, R.D.: Randomized primal-dual analysis of ranking for online bipartite matching. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, June 2013. https://doi.org/10.1137/1.9781611973105.7
Feldman, J., Mehta, A., Mirrokni, V., Muthukrishnan, S.: Online stochastic matching: beating 1–1/e. In: 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 117–126 (2009). https://doi.org/10.1109/FOCS.2009.72
Goel, G., Mehta, A.: Online budgeted matching in random input models with applications to adwords. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, pp. 982–991. Society for Industrial and Applied Mathematics (2008)
Huang, Z., Shu, X.: Online stochastic matching, poisson arrivals, and the natural linear program (2021). arXiv:2103.13024
Huang, Z., Tang, Z.G., Wu, X., Zhang, Y.: Online vertex-weighted bipartite matching: beating \(1-1/e\) with random arrivals. ACM Trans. Algorithms 15(3) (2019). https://doi.org/10.1145/3326169
Jaillet, P., Xin, L.: Online stochastic matching: new algorithms with better bounds. Math. Oper. Res. 39(3), 624–646 (2014). https://doi.org/10.1287/moor.2013.0621
Karande, C., Mehta, A., Tripathi, P.: Online bipartite matching with unknown distributions. In: Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, STOC 2011, pp. 587–596 (2011). https://doi.org/10.1145/1993636.1993715
Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing - STOC 90 (1990). https://doi.org/10.1145/100216.100262
Mahdian, M., Yan, Q.: Online bipartite matching with random arrivals. In: Proceedings of the 43rd Annual ACM Symposium on Theory of Computing - STOC 11 (2011). https://doi.org/10.1145/1993636.1993716
Manshadi, V.H., Gharan, S.O., Saberi, A.: Online stochastic matching: online actions based on offline statistics. Math. Oper. Res. 37(4), 559–573 (2012). https://doi.org/10.1287/moor.1120.0551
Manshadi, V.H., Gharan, S.O., Saberi, A.: Online stochastic matching: online actions based on offline statistics. Math. Oper. Res. 37, 559–573 (2012)
Mehta, A.: Online matching and ad allocation. Found. Trends Theor. Comput. Sci. 8(4), 265–368 (2013). https://doi.org/10.1561/0400000057
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Jin, B., Williamson, D.P. (2022). Improved Analysis of RANKING for Online Vertex-Weighted Bipartite Matching in the Random Order Model. In: Feldman, M., Fu, H., Talgam-Cohen, I. (eds) Web and Internet Economics. WINE 2021. Lecture Notes in Computer Science(), vol 13112. Springer, Cham. https://doi.org/10.1007/978-3-030-94676-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-030-94676-0_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-94675-3
Online ISBN: 978-3-030-94676-0
eBook Packages: Computer ScienceComputer Science (R0)