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# Metatheorems for Dynamic Weighted Matching

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## Cite As

Daniel Stubbs and Virginia Vassilevska Williams. Metatheorems for Dynamic Weighted Matching. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ITCS.2017.58

## Abstract

We consider the maximum weight matching (MWM) problem in dynamic graphs. We provide two reductions. The first reduces the dynamic MWM problem on m-edge, n-node graphs with weights bounded by N to the problem with weights bounded by (n/eps)^2, so that if the MWM problem can be alpha-approximated with update time t(m,n,N), then it can also be (1+eps)alpha-approximated with update time O(t(m,n,(n/eps)^2)log^2 n+log n loglog N)). The second reduction reduces the dynamic MWM problem to the dynamic maximum cardinality matching (MCM) problem in which the graph is unweighted. This reduction shows that if there is an \alpha-approximation algorithm for MCM with update time t(m,n) in m-edge n-node graphs, then there is also a (2+eps)alpha-approximation algorithm for MWM with update time O(t(m,n)eps^{-2}log^2 N). We also obtain better bounds in our reductions if the ratio between the largest and the smallest edge weight is small. Combined with recent work on MCM, these two reductions substantially improve upon the state-of-the-art of dynamic MWM algorithms.
##### Keywords
• dynamic algorithms
• maximum matching
• maximum weight matching

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## References

1. Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 434-443, 2014.
2. A. Anand, S. Baswana, M. Gupta, and S. Sen. Maintaining approximate maximum weighted matching in fully dynamic graphs. In FSTTCS, pages 257-266, 2012.
3. S. Baswana, M. Gupta, and S. Sen. Fully dynamic maximal matching in O(log n) update time. In FOCS, pages 383-392, 2011.
4. Aaron Bernstein and Cliff Stein. Fully dynamic matching in bipartite graphs. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 167-179, 2015.
5. Aaron Bernstein and Cliff Stein. Faster fully dynamic matchings with small approximation ratios. In In Proc. SODA, page to appear, 2016.
6. Sayan Bhattacharya, Monika Henzinger, and Giuseppe F. Italiano. Deterministic fully dynamic data structures for vertex cover and matching. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 785-804, 2015.
7. Sayan Bhattacharya, Monika Henzinger, and Danupon Nanangkai. New deterministic approximation algorithms for fully dynamic matching. In Proc. STOC, page to appear, 2016.
8. Michael Crouch and Daniel Stubbs. Improved streaming algorithms for weighted matching, via unweighted matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, September 4-6, 2014, Barcelona, Spain, pages 96-104, 2014.
9. H. Gabow. A scaling algorithm for weighted matching on general graphs. In Prof. FOCS, pages 90-100, 1985.
10. H. N. Gabow and R. E. Tarjan. Faster scaling algorithms for general graph-matching problems. J. ACM, 38(4):815-853, 1991.
11. M. Gupta and R. Peng. Fully dynamic (1+ε)-approximate matchings. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 548-557, 2013.
12. N. J. A. Harvey. Algebraic structures and algorithms for matching and matroid problems. In Proc. FOCS, volume 47, pages 531-542, 2006.
13. J. Hopcroft and R. Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing, 2(4):225-231, 1973.
14. Jonathan A. Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 217-226, 2014.
15. H. W. Kuhn. The hungarian method for the assignment problem. Naval Research Logistics Quarterly, 2:83-97, 1955.
16. François Le Gall. Powers of tensors and fast matrix multiplication. In International Symposium on Symbolic and Algebraic Computation, ISSAC '14, Kobe, Japan, July 23-25, 2014, pages 296-303, 2014.
17. A. Madry. Navigating central path with electrical flows: from flows to matchings, and back. In Proc. FOCS, 2013.
18. Silvio Micali and Vijay V. Vazirani. An o(sqrt(|v|) |e|) algorithm for finding maximum matching in general graphs. In 21st Annual Symposium on Foundations of Computer Science, Syracuse, New York, USA, 13-15 October 1980, pages 17-27, 1980.
19. M. Mucha and P. Sankowski. Maximum matchings via gaussian elimination. In Proc. FOCS, volume 45, pages 248-255, 2004.
20. K. Mulmuley, U. V. Vazirani, and V. V. Vazirani. Matching is as easy as matrix inversion. In Proc. STOC, volume 19, pages 345-354, 1987.
21. O. Neiman and S. Solomon. Simple deterministic algorithms for fully dynamic maximal matching. In Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC '13, pages 745-754, 2013.
22. Krzysztof Onak and Ronitt Rubinfeld. Maintaining a large matching and a small vertex cover. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 457-464, 2010.
23. M. O. Rabin and V. V. Vazirani. Maximum matchings in general graphs through randomization. J. Algorithms, 10(4):557-567, 1989.
24. P. Sankowski. Faster dynamic matchings and vertex connectivity. In Proc. SODA, pages 118-126, 2007.
25. P. Sankowski. Maximum weight bipartite matching in matrix multiplication time. Theor. Comput. Sci., 410(44):4480-4488, 2009.
26. V. Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd. In Proc. STOC, pages 887-898, 2012.
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