skip to main content
10.1145/1143997.1144095acmconferencesArticle/Chapter ViewAbstractPublication PagesgeccoConference Proceedingsconference-collections
Article

Maximum cardinality matchings on trees by randomized local search

Published: 08 July 2006 Publication History

Abstract

To understand the working principles of randomized search heuristics like evolutionary algorithms they are analyzed on optimization problems whose structure is well-studied. The idea is to investigate when it is possible to simulate clever optimization techniques for combinatorial optimization problems by random search. The maximum matching problem is well suited for this approach since long augmenting paths do not allow immediate improvements by local changes. It is known that randomized search heuristics like simulated annealing, the Metropolis algorithm, the (1+1) EA and randomized local search efficiently approximate maximum matchings for any graph; however, there are graphs where they fail to find maximum matchings in polynomial time. In this paper, we examine randomized local search (RLS) for graphs whose structure is simple. We show that RLS finds maximum matchings on trees in expected polynomial time.

References

[1]
C. Berge. Two theorems in graph theory. Proceedings of the National Academy of Sciences of the United States of America, 43:842--844, 1957.
[2]
P. Briest, D. Brockhoff, B. Degener, M. Englert, C. Gunia, O. Heering, T. Jansen, M. Leifhelm, K. Plociennik, H. Röglin, A. Schweer, D. Sudholt, S. Tannenbaum, and I. Wegener. Experimental supplements to the theoretical analysis of EAs on problems from combinatorial optimization. In Proceedings of the 8th Conference on Parallel Problem Solving from Nature (PPSN '04), volume 3242 of Lecture Notes in Computer Science, pages 21--30. Springer, 2004.
[3]
J. Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17:449--467, 1965.
[4]
O. Giel and I. Wegener. Evolutionary algorithms and the maximum matching problem. In Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science (STACS '03), volume 2607 of LNCS, pages 415--426. Springer, 2003.
[5]
J. He and X. Yao. Time complexity analysis of an evolutionary algorithm for finding nearly maximum cardinality matching. Journal of Computer Science and Technology, 19(4):450--458, 2004.
[6]
J. E. Hopcroft and R. M. Karp. An n5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing, 2(4):225--231, 1973.
[7]
M. Jerrum. Large cliques elude the Metropolis process. Random Structures and Algorithms, 3(4):347--359, 1992.
[8]
M. Jerrum and G. B. Sorkin. The Metropolis algorithm for graph bisection. Discrete Applied Mathematics, 82:155--175, 1998.
[9]
S. Micali and V. V. Vazirani. An O(√|V| ⋅ |E|) algorithm for finding maximum matching in general graphs. In Proceedings of the 21st Annual Symposium on Foundations of Computer Science (FOCS '80), pages 17--27. IEEE, 1980.
[10]
F. Neumann and I. Wegener. Randomized local search, evolutionary algorithms, and the minimum spanning tree problem. In Proceedings of the 6th Genetic and Evolutionary Computation Conference (GECCO '04), volume 3102 of LNCS, pages 713--724. Springer, 2004.
[11]
G. Sasaki. The effect of the density of states on the Metropolis algorithm. Information Processing Letters, 37(3):159--163, 1991.
[12]
G. H. Sasaki and B. Hajek. The time complexity of maximum matching by simulated annealing. Journal of the ACM, 35:387--403, 1988.
[13]
J. Scharnow, K. Tinnefeld, and I. Wegener. Fitness landscapes based on sorting and shortest paths problems. In Proceedings of the 7th Conference on Parallel Problem Solving from Nature (PPSN '02), volume 2439 of LNCS, pages 54--63. Springer, 2002.
[14]
V. V. Vazirani. A theory of alternating paths and blossoms for proving correctness of the O(√V E) maximum matching algorithm. Combinatorica, 14:71--109, 1994.
[15]
I. Wegener. Simulated annealing beats Metropolis in combinatorial optimization. In Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP '05), volume 3580 of LNCS, pages 589--601, 2005.
[16]
C. Witt. Worst-case and average-case approximations by simple randomized search heuristics. In Proceedings of the 22nd Annual Symposium on Theoretical Aspects of Computer Science (STACS '05), volume 3404 of LNCS, pages 44--56. Springer, 2005.

Cited By

View all
  • (2018)Analyzing evolutionary optimization in noisy environmentsEvolutionary Computation10.1162/evco_a_0017026:1(1-41)Online publication date: 1-Mar-2018
  • (2018)Performance Analysis of Evolutionary Optimization for the Bank Account Location ProblemIEEE Access10.1109/ACCESS.2017.27791546(17756-17767)Online publication date: 2018
  • (2013)A runtime analysis of simple hyper-heuristicsProceedings of the twelfth workshop on Foundations of genetic algorithms XII10.1145/2460239.2460249(97-104)Online publication date: 16-Jan-2013
  • Show More Cited By

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences
GECCO '06: Proceedings of the 8th annual conference on Genetic and evolutionary computation
July 2006
2004 pages
ISBN:1595931864
DOI:10.1145/1143997
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 08 July 2006

Permissions

Request permissions for this article.

Check for updates

Author Tags

  1. evolutionary algorithms
  2. maximum cardinality matchings
  3. randomized local search
  4. runtime analysis

Qualifiers

  • Article

Conference

GECCO06
Sponsor:
GECCO06: Genetic and Evolutionary Computation Conference
July 8 - 12, 2006
Washington, Seattle, USA

Acceptance Rates

GECCO '06 Paper Acceptance Rate 205 of 446 submissions, 46%;
Overall Acceptance Rate 1,669 of 4,410 submissions, 38%

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)2
  • Downloads (Last 6 weeks)1
Reflects downloads up to 17 Jan 2025

Other Metrics

Citations

Cited By

View all
  • (2018)Analyzing evolutionary optimization in noisy environmentsEvolutionary Computation10.1162/evco_a_0017026:1(1-41)Online publication date: 1-Mar-2018
  • (2018)Performance Analysis of Evolutionary Optimization for the Bank Account Location ProblemIEEE Access10.1109/ACCESS.2017.27791546(17756-17767)Online publication date: 2018
  • (2013)A runtime analysis of simple hyper-heuristicsProceedings of the twelfth workshop on Foundations of genetic algorithms XII10.1145/2460239.2460249(97-104)Online publication date: 16-Jan-2013
  • (2010)Analysis of Evolutionary Algorithms for the Longest Common Subsequence ProblemAlgorithmica10.5555/3118215.311826857:1(170-186)Online publication date: 1-May-2010
  • (2010)Evolutionary Algorithms and Matroid Optimization ProblemsAlgorithmica10.5555/3118215.311826557:1(187-206)Online publication date: 1-May-2010
  • (2008)Simulated annealing, its parameter settings and the longest common subsequence problemProceedings of the 10th annual conference on Genetic and evolutionary computation10.1145/1389095.1389253(803-810)Online publication date: 13-Jul-2008
  • (2008)Evolutionary Algorithms and Matroid Optimization ProblemsAlgorithmica10.1007/s00453-008-9253-457:1(187-206)Online publication date: 2-Dec-2008
  • (2008)Analysis of Evolutionary Algorithms for the Longest Common Subsequence ProblemAlgorithmica10.1007/s00453-008-9243-657:1(170-186)Online publication date: 25-Oct-2008
  • (2007)Evolutionary algorithms and matroid optimization problemsProceedings of the 9th annual conference on Genetic and evolutionary computation10.1145/1276958.1277149(947-954)Online publication date: 7-Jul-2007
  • (2007)Analysis of evolutionary algorithms for the longest common subsequence problemProceedings of the 9th annual conference on Genetic and evolutionary computation10.1145/1276958.1277148(939-946)Online publication date: 7-Jul-2007
  • Show More Cited By

View Options

Login options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media