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
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)