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Scaling Algorithms for Weighted Matching in General Graphs

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Hsin-Hao SuAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 14, Issue 1

Article No.: 8, Pages 1 - 35

https://doi.org/10.1145/3155301

Published: 03 January 2018 Publication History

Abstract

We present a new scaling algorithm for maximum (or minimum) weight perfect matching on general, edge weighted graphs. Our algorithm runs in O(m√nlog(nN)) time, O(m√ n) per scale, which matches the running time of the best cardinality matching algorithms on sparse graphs [16, 20, 36, 37]. Here, m,n, and N bound the number of edges, vertices, and magnitude, respectively, of any integer edge weight. Our result improves on a 25-year-old algorithm of Gabow and Tarjan, which runs in O(m√nlog nα (m,n) log(nN)) time.

References

[1]

G. Birkhoff. 1946. Tres observaciones sobre el elgebra lineal. Universidad Nacional de Tucuman, Revista A 5, 1--2, 147--151.

[2]

M. B. Cohen, A. Madry, P. Sankowski, and A. Vladu. 2017. Negative-weight shortest paths and unit capacity minimum cost flow in time. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’17). 752--771.

Digital Library

[3]

W. H. Cunningham and A. B. Marsh, III. 1978. A primal algorithm for optimum matching. Math. Program. Study 8, 50--72.

[4]

M. Cygan, H. N. Gabow, and P. Sankowski. 2015. Algorithmic applications of Baur-Strassen’s theorem: Shortest cycles, diameter, and matchings. J. ACM 62, 4, 28.

Digital Library

[5]

R. Duan and S. Pettie. 2014. Linear-time approximation for maximum weight matching. J. ACM 61, 1, 1.

Digital Library

[6]

R. Duan and H.-H. Su. 2012. A scaling algorithm for maximum weight matching in bipartite graphs. In Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms (SODA’12). 1413--1424.

Digital Library

[7]

J. Edmonds. 1965. Maximum matching and a polyhedron with -vertices. J. Res. Nat. Bur. Stand. Sect. B 69B, 125--130.

[8]

J. Edmonds 1965. Paths, trees, and flowers. Can. J. Math. 17, 449--467.

[9]

J. Edmonds and R. M. Karp 1972. Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19, 2, 248--264.

Digital Library

[10]

M. L. Fredman and R. E. Tarjan. 1987. Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34, 3, 596--615.

Digital Library

[11]

M. L. Fredman and D. E. Willard. 1994. Trans-dichotomous algorithms for minimum spanning trees and shortest paths. J. Comput. Syst. Sci. 48, 3, 533--551.

Digital Library

[12]

H. N. Gabow. 1976. An efficient implementation of Edmonds’ algorithm for maximum matching on graphs. J. ACM 23, 221--234.

Digital Library

[13]

H. N. Gabow. 1985. A scaling algorithm for weighted matching on general graphs. In Proceedings of the 26th Annual Symposium on Foundations of Computer Science (FOCS’85). 90--100.

Digital Library

[14]

H. N. Gabow. 1990. Data structures for weighted matching and nearest common ancestors with linking. In Proceedings of the 1st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’90). 434--443.

Digital Library

[15]

H. N. Gabow. 2016. Data structures for weighted matching and extensions to -matching and -factors. CoRR, abs/1611.07541.

[16]

H. N. Gabow. 2017. The weighted matching approach to maximum cardinality matching. Fundam. Inform. 154, 1--4, 109--130.

[17]

H. N. Gabow, Z. Galil, and T. H. Spencer. 1989. Efficient implementation of graph algorithms using contraction. J. ACM 36, 3, 540--572.

Digital Library

[18]

H. N. Gabow and R. E. Tarjan. 1985. A linear-time algorithm for a special case of disjoint set union. J. Comput. Syst. Sci. 30, 2, 209--221.

[19]

H. N. Gabow and R. E. Tarjan. 1989. Faster scaling algorithms for network problems. SIAM J. Comput. 18, 5, 1013--1036.

Digital Library

[20]

H. N. Gabow and R. E. Tarjan. 1991. Faster scaling algorithms for general graph-matching problems. J. ACM 38, 4, 815--853.

Digital Library

[21]

Z. Galil, S. Micali, and H. N. Gabow. 1986. An algorithm for finding a maximal weighted matching in general graphs. SIAM J. Comput. 15, 1, 120--130.

Digital Library

[22]

A. V. Goldberg and R. Kennedy. 1997. Global price updates help. SIAM J. Discr. Math. 10, 4, 551--572.

Digital Library

[23]

Y. Han. 2002. Deterministic sorting in time and linear space. In Proceedings of the 34th ACM Symposium on Theory of Computers (STOC’02). ACM Press, 602--608.

Digital Library

[24]

Y. Han and M. Thorup. 2002. Integer sorting in O(n√log log n) expected time and linear space. In Proceedings of the 43rd Symposium on Foundations of Computer Science (FOCS’02). 135--144.

Digital Library

[25]

C.-C. Huang and T. Kavitha. 2012. Efficient algorithms for maximum weight matchings in general graphs with small edge weights. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’12). 1400--1412.

Digital Library

[26]

M.-Y. Kao, T. W. Lam, W.-K. Sung, and H.-F. Ting. 2001. A decomposition theorem for maximum weight bipartite matchings. SIAM J. Comput. 31, 1, 18--26.

Digital Library

[27]

A. V. Karzanov. 1976. Efficient implementations of Edmonds’ algorithms for finding matchings with maximum cardinality and maximum weight. In Studies in Discrete Optimization, A. A. Fridman (Eds.). Nauka, Moscow, 306--327.

[28]

E. Lawler. 1976. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart 8 Winston, New York.

[29]

S. Micali and V. Vazirani. 1980. An O(√|V| ċ |E|) algorithm for finding maximum matching in general graphs. In Proceedings of the 21st IEEE Symposium on Foundations of Computer Science (FOCS’80). 17--27.

Digital Library

[30]

J. B. Orlin and R. K. Ahuja. 1992. New scaling algorithms for the assignment and minimum mean cycle problems. Math. Program. 54, 41--56.

Digital Library

[31]

S. Pettie. 2012. A simple reduction from maximum weight matching to maximum cardinality matching. Inf. Process. Lett. 112, 23, 893--898.

Digital Library

[32]

S. Pettie. 2015. Sensitivity analysis of minimum spanning trees in sub-inverse-Ackermann time. J. Graph Algor. Appl. 19, 1, 375--391.

[33]

M. Thorup. 1999. Undirected single-source shortest paths with positive integer weights in linear time. J. ACM 46, 3, 362--394.

Digital Library

[34]

M. Thorup. 2003. Integer priority queues with decrease key in constant time and the single source shortest paths problem. In Proceedings of the 35th ACM Symposium on Theory of Computing (STOC’03). 149--158.

Digital Library

[35]

M. Thorup. 2007. Equivalence between priority queues and sorting. J. ACM 54, 6.

Digital Library

[36]

V. V. Vazirani. 2012. An improved definition of blossoms and a simpler proof of the MV matching algorithm. CoRR, abs/1210.4594.

[37]

V. V. Vazirani. 2014. A proof of the MV matching algorithm. Unpublished manuscript.

[38]

J. von Neumann. 1953. A certain zero-sum two-person game equivalent to the optimal assignment problem. In Contributions to the Theory of Games, vol. II, H. W. Kuhn and A. W. Tucker (Eds.). Princeton University Press, 5--12.

Cited By

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https://doi.org/10.1016/j.tcs.2024.114903
Gabow H(2024)Maximum Cardinality f-Matching in Time O(n2/3m)ACM Transactions on Algorithms10.1145/369666821:1(1-28)Online publication date: 23-Sep-2024
https://dl.acm.org/doi/10.1145/3696668
Merino AMütze T(2024)Traversing Combinatorial 0/1-Polytopes via OptimizationSIAM Journal on Computing10.1137/23M161201953:5(1257-1292)Online publication date: 9-Sep-2024
https://doi.org/10.1137/23M1612019
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Index Terms

Scaling Algorithms for Weighted Matching in General Graphs
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
      2. Matchings and factors
2. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis

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cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 14, Issue 1

January 2018

269 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3171590

Editor:
Aravind Srinivasan
University of Maryland, USA

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Copyright © 2018 ACM.

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Publication History

Published: 03 January 2018

Accepted: 01 October 2017

Revised: 01 October 2017

Received: 01 February 2017

Published in TALG Volume 14, Issue 1

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Gabow H(2024)Maximum Cardinality f-Matching in Time O(n2/3m)ACM Transactions on Algorithms10.1145/369666821:1(1-28)Online publication date: 23-Sep-2024
https://dl.acm.org/doi/10.1145/3696668
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https://doi.org/10.1137/23M1612019
Schlotter I(2024)Recognizing when a preference system is close to admitting a master listTheoretical Computer Science10.1016/j.tcs.2024.114445994(114445)Online publication date: May-2024
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Boehmer NHeeger K(2024)Adapting Stable Matchings to Forced and Forbidden PairsJournal of Computer and System Sciences10.1016/j.jcss.2024.103579(103579)Online publication date: Aug-2024
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Richer MKim TAyers P(2024)Graphical Approach to Interpreting and Efficiently Evaluating Geminal WavefunctionsInternational Journal of Quantum Chemistry10.1002/qua.70000125:1Online publication date: 20-Dec-2024
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Brand JChen LKyng RLiu YPeng RGutenberg MSachdeva SSidford A(2023)A Deterministic Almost-Linear Time Algorithm for Minimum-Cost Flow2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00037(503-514)Online publication date: 6-Nov-2023
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Subramani KWojciechowski P(2023)Constrained read-once refutations in UTVPI constraint systems: A parallel perspectiveMathematical Structures in Computer Science10.1017/S0960129523000300(1-17)Online publication date: 11-Sep-2023
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Gabow H(2023)A Weight-Scaling Algorithm for f-Factors of MultigraphsAlgorithmica10.1007/s00453-023-01127-x85:10(3214-3289)Online publication date: 18-May-2023
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