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Adjacency list matchings: an ideal genotype for cycle covers

Published: 07 July 2007 Publication History

Abstract

We propose and analyze a novel genotype to represent walk and cycle covers in graphs, namely matchings in the adjacency lists. This representation admits the natural mutation operator of adding a random match and possibly also matching the former partners.
To demonstrate the strength of this set-up, we use it to build a simple (1+1) evolutionary algorithm for the problem of finding an Eulerian cycle in a graph. We analyze several natural variants that stem from different ways to randomly choose the new match.
Among other insight, we exhibit a (1+1) evolutionary algorithm that computes an Euler tour in a graph with $m$ edges in expected optimization time Θ(m log m). This significantly improves the previous best evolutionary solution having expected optimization time Θ(m2 log m) in the worst-case, but also compares nicely with the runtime of an optimal classical algorithm which is of order Θ(m). A simple coupon collector argument indicates that our optimization time is asymptotically optimal for any randomized search heuristic.

References

[1]
B. Doerr, N. Hebbinghaus, and F. Neumann. Speeding up evolutionary algorithms through restricted mutation operators. In Proc. of the 9th International Conference on Parallel Problem Solving From Nature (PPSN), volume 4193 of LNCS, pages 978--987. Springer, 2006.
[2]
B. Doerr, C. Klein, and T. Storch. Faster evolutionary algorithms by superior graph representation. In Proc. of the 2007 IEEE Symposium on Foundations of Computational Intelligence (FOCI), pages 245--250. IEEE, 2007.
[3]
J. Edmonds and E. L. Johnson. Matching, euler tours and the chinese postman. Mathematical Programming, 5:88--124, 1973.
[4]
L. Euler. Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae, 8:128--140, 1741.
[5]
C. Hierholzer. Ueber die M.oglichkeit, einen Linenzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Mathematische Annalen, 6:30--32, 1873.
[6]
F. Neumann. Expected runtimes of evolutionary algorithms for the Eulerian cycle problem. In Proc. of the 2004 IEEE Congress on Evolutionary Computation (CEC), pages 904--910. IEEE, 2004.

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    cover image ACM Conferences
    GECCO '07: Proceedings of the 9th annual conference on Genetic and evolutionary computation
    July 2007
    2313 pages
    ISBN:9781595936974
    DOI:10.1145/1276958
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    Published: 07 July 2007

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    Author Tags

    1. Euler tour
    2. cycle cover
    3. evolutionary algorithm
    4. randomized local
    5. runtime analysis
    6. search

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    GECCO '07 Paper Acceptance Rate 266 of 577 submissions, 46%;
    Overall Acceptance Rate 1,669 of 4,410 submissions, 38%

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    • (2023)Curing ill-Conditionality via Representation-Agnostic Distance-Driven PerturbationsProceedings of the 17th ACM/SIGEVO Conference on Foundations of Genetic Algorithms10.1145/3594805.3607138(15-26)Online publication date: 30-Aug-2023
    • (2023)A Gentle Introduction to Theory (for Non-Theoreticians)Proceedings of the Companion Conference on Genetic and Evolutionary Computation10.1145/3583133.3595042(946-975)Online publication date: 15-Jul-2023
    • (2023)Runtime Analysis for Permutation-based Evolutionary AlgorithmsAlgorithmica10.1007/s00453-023-01146-886:1(90-129)Online publication date: 26-Jul-2023
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    • (2021)A gentle introduction to theory (for non-theoreticians)Proceedings of the Genetic and Evolutionary Computation Conference Companion10.1145/3449726.3461406(369-398)Online publication date: 7-Jul-2021
    • (2020)A gentle introduction to theory (for non-theoreticians)Proceedings of the 2020 Genetic and Evolutionary Computation Conference Companion10.1145/3377929.3389889(373-403)Online publication date: 8-Jul-2020
    • (2019)Theory for non-theoreticiansProceedings of the Genetic and Evolutionary Computation Conference Companion10.1145/3319619.3323373(523-549)Online publication date: 13-Jul-2019
    • (2018)Theory for non-theoreticiansProceedings of the Genetic and Evolutionary Computation Conference Companion10.1145/3205651.3207889(389-414)Online publication date: 6-Jul-2018
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