Dynamic Matching Algorithms Under Vertex Updates
Dynamic graph matching algorithms have been extensively studied, but mostly under edge updates. This paper concerns dynamic matching algorithms under vertex updates, where in each update step a single vertex is either inserted or deleted along with its incident edges.
A basic setting arising in online algorithms and studied by Bosek et al. [FOCS'14] and Bernstein et al. [SODA'18] is that of dynamic approximate maximum cardinality matching (MCM) in bipartite graphs in which one side is fixed and vertices on the other side either arrive or depart via vertex updates. In the BASIC-incremental setting, vertices only arrive, while in the BASIC-decremental setting vertices only depart. When vertices can both arrive and depart, we have the BASIC-dynamic setting. In this paper we also consider the setting in which both sides of the bipartite graph are dynamic. We call this the MEDIUM-dynamic setting, and MEDIUM-decremental is the restriction when vertices can only depart. The GENERAL-dynamic setting is when the graph is not necessarily bipartite and the vertices can both depart and arrive.
Denote by K the total number of edges inserted and deleted to and from the graph throughout the entire update sequence. A well-studied measure, the recourse of a dynamic matching algorithm is the number of changes made to the matching per step. We largely focus on Maximal Matching (MM) which is a 2-approximation to the MCM. Our main results are as follows.
- In the BASIC-dynamic setting, there is a straightforward algorithm for maintaining a MM, with a total runtime of O(K) and constant worst-case recourse. In fact, this algorithm never removes an edge from the matching; we refer to such an algorithm as irrevocable.
- For the MEDIUM-dynamic setting we give a strong conditional lower bound that even holds in the MEDIUM-decremental setting: if for any fixed η > 0, there is an irrevocable decremental MM algorithm with a total runtime of O(K ⋅ n^{1-η}), this would refute the OMv conjecture; a similar (but weaker) hardness result can be achieved via a reduction from the Triangle Detection conjecture.
- Next, we consider the GENERAL-dynamic setting, and design an MM algorithm with a total runtime of O(K) and constant worst-case recourse. We achieve this result via a 1-revocable algorithm, which may remove just one edge per update step. As argued above, an irrevocable algorithm with such a runtime is not likely to exist.
- Finally, back to the BASIC-dynamic setting, we present an algorithm with a total runtime of O(K), which provides an (e/(e-1))-approximation to the MCM.
To this end, we build on the classic "ranking" online algorithm by Karp et al. [STOC'90]. Beyond the results, our work draws connections between the areas of dynamic graph algorithms and online algorithms, and it proposes several open questions that seem to be overlooked thus far.
maximal matching
approximate matching
dynamic algorithm
vertex updates
Theory of computation~Dynamic graph algorithms
96:1-96:24
Regular Paper
Hung
Le
Hung Le
University of Massachusetts, Amherst, MA, USA
Supported by National Science Foundation under Grant No. CCF-2121952.
Lazar
Milenković
Lazar Milenković
Tel Aviv University, Israel
Partially supported by the Israel Science Foundation (ISF) grant No.1991/1, and by a grant from the United States-Israel Binational Science Foundation (BSF), Israel, and the United States National Science Foundation (NSF).
Shay
Solomon
Shay Solomon
Tel Aviv University, Israel
Supported by the Israel Science Foundation (ISF) grant No.1991/1, and by a grant from the United States-Israel Binational Science Foundation (BSF), Israel, and the United States National Science Foundation (NSF).
Virginia
Vassilevska Williams
Virginia Vassilevska Williams
MIT, Cambridge, MA, USA
Supported in part by NSF CAREER Award 1651838, NSF Grants CCF-1909429 and CCF- 2129139, BSF grants 2016365 and 2020356, a Google Research Fellowship and a Sloan Research Fellowship.
10.4230/LIPIcs.ITCS.2022.96
Ittai Abraham, David Durfee, Ioannis Koutis, Sebastian Krinninger, and Richard Peng. On fully dynamic graph sparsifiers. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 335-344. IEEE Computer Society, 2016.
S. Baswana, M. Gupta, and S. Sen. Fully dynamic maximal matching in O(log n) update time. In Proceedings of the 52nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, October 23-25, 2011, pages 383-392, 2011. See also the SICOMP'15 version, and the subsequent erratum.
Soheil Behnezhad, Jakub Lacki, and Vahab S. Mirrokni. Fully dynamic matching: Beating 2-approximation in Δ^ε update time. In Shuchi Chawla, editor, Proc. 51th SODA, pages 2492-2508, 2020.
Aaron Bernstein, Jacob Holm, and Eva Rotenberg. Online bipartite matching with amortized replacements. In SODA, pages 947-959. SIAM, 2018.
Aaron Bernstein, Tsvi Kopelowitz, Seth Pettie, Ely Porat, and Clifford Stein. Simultaneously load balancing for every p-norm, with reassignments. In Proc. 8th ITCS, pages 51:1-51:14, 2017.
Aaron Bernstein and Cliff Stein. Fully dynamic matching in bipartite graphs. In Proc. 42nd ICALP, pages 167-179, 2015.
Aaron Bernstein and Cliff Stein. Faster fully dynamic matchings with small approximation ratios. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 692-711, 2016.
Sayan Bhattacharya, Monika Henzinger, and Danupon Nanongkai. New deterministic approximation algorithms for fully dynamic matching. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 398-411, 2016.
Sayan Bhattacharya and Peter Kiss. Deterministic rounding of dynamic fractional matchings. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, Proc. 48th ICALP, volume 198, pages 27:1-27:14, 2021.
Benjamin E. Birnbaum and Claire Mathieu. On-line bipartite matching made simple. SIGACT News, 39(1):80-87, 2008.
Bartlomiej Bosek, Dariusz Leniowski, Piotr Sankowski, and Anna Zych. Online bipartite matching in offline time. In Proc. 55th FOCS, pages 384-393, 2014.
Bartlomiej Bosek, Dariusz Leniowski, Piotr Sankowski, and Anna Zych. Shortest augmenting paths for online matchings on trees. In Proc. of 13th WAOA, pages 59-71, 2015.
Bartłomiej Bosek, Dariusz Leniowski, Piotr Sankowski, and Anna Zych-Pawlewicz. A tight bound for shortest augmenting paths on trees. In Proc. 13th LATIN, pages 201-216, 2018.
Kamalika Chaudhuri, Constantinos Daskalakis, Robert D. Kleinberg, and Henry Lin. Online bipartite perfect matching with augmentations. In Proc. of 28th INFOCOM, pages 1044-1052, 2009.
Gagan Goel and Aranyak Mehta. Online budgeted matching in random input models with applications to adwords. In SODA, pages 982-991. SIAM, 2008.
Fabrizio Grandoni, Stefano Leonardi, Piotr Sankowski, Chris Schwiegelshohn, and Shay Solomon. (1 + ε)-approximate incremental matching in constant deterministic amortized time. In Timothy M. Chan, editor, Proc. 50th SODA, pages 1886-1898, 2019.
Fabrizio Grandoni, Chris Schwiegelshohn, Shay Solomon, and Amitai Uzrad. Maintaining an EDCS in general graphs: Simpler, density-sensitive and with worst-case time bounds. CoRR, abs/2108.08825, 2021.
Edward F. Grove, Ming-Yang Kao, P. Krishnan, and Jeffrey Scott Vitter. Online perfect matching and mobile computing. In Proc. of 45th Wads, pages 194-205, 1995.
M. Gupta and R. Peng. Fully dynamic (1+ε)-approximate matchings. In Proceedings of the 54th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2013, Berkeley, CA, USA, October 26-29, 2013, pages 548-557, 2013.
Meng He, Ganggui Tang, and Norbert Zeh. Orienting dynamic graphs, with applications to maximal matchings and adjacency queries. In ISAAC, volume 8889 of Lecture Notes in Computer Science, pages 128-140. Springer, 2014.
Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Proceedings of the 47th Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 21-30, 2015.
John E. Hopcroft and Richard M. Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput., 2(4):225-231, 1973. URL: https://doi.org/10.1137/0202019.
https://doi.org/10.1137/0202019
Zhiyi Huang, Ning Kang, Zhihao Gavin Tang, Xiaowei Wu, Yuhao Zhang, and Xue Zhu. How to match when all vertices arrive online. In STOC, pages 17-29. ACM, 2018.
Z. Ivković and E. L. Lloyd. Fully dynamic maintenance of vertex cover. In Proceedings of the 19th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 1993, Utrecht, The Netherlands, June 16-18, 1993, pages 99-111, 1993.
Richard M. Karp, Umesh V. Vazirani, and Vijay V. Vazirani. An optimal algorithm for on-line bipartite matching. In Proc. 22nd STOC, pages 352-358, 1990.
Lazar Milenkovic and Shay Solomon. A unified sparsification approach for matching problems in graphs of bounded neighborhood independence. In Christian Scheideler and Michael Spear, editors, Proc. of 32nd SPAA, pages 395-406. ACM, 2020.
Ofer Neiman and Shay Solomon. Simple deterministic algorithms for fully dynamic maximal matching. In Proceedings of the 45th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2013, Palo Alto, CA, USA, June 1-4, 2013, pages 745-754, 2013.
D. Peleg and S. Solomon. Dynamic (1 + ε)-approximate matchings: A density-sensitive approach. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, 2016.
Yongho Shin, Kangsan Kim, Seungmin Lee, and Hyung-Chan An. Online graph matching problems with a worst-case reassignment budget. CoRR, abs/2003.05175, 2020.
Noam Solomon and Shay Solomon. A generalized matching reconfiguration problem. In ITCS, volume 185 of LIPIcs, pages 57:1-57:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021.
S. Solomon. Fully dynamic maximal matching in constant update time. In Proceedings of the 57th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2016, New Brunswick, NJ, USA, October 9-11, 2016, pages 325-334, 2016.
Jan van den Brand, Danupon Nanongkai, and Thatchaphol Saranurak. Dynamic matrix inverse: Improved algorithms and matching conditional lower bounds. In FOCS, pages 456-480. IEEE Computer Society, 2019.
David Wajc. Rounding dynamic matchings against an adaptive adversary. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proc. 52nd STOC, pages 194-207, 2020.
Hung Le, Lazar Milenković, Shay Solomon, and Virginia Vassilevska Williams
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode