research-article

Evolving labelings of graceful graphs

Authors:

Andrew M. SuttonAuthors Info & Claims

GECCO '22: Proceedings of the Genetic and Evolutionary Computation Conference

Pages 195 - 203

https://doi.org/10.1145/3512290.3528855

Published: 08 July 2022 Publication History

Abstract

A graceful labeling of a graph G = (V, E) is an assignment of labels to the vertices V of G subject to constraints arising from the structure of the graph. A graph is called graceful if it admits a graceful labeling. As a combinatorial problem, it has applications in coding theory, communications networks, and optimizing circuit layouts. Several different approaches, both heuristic and complete, for finding graceful labelings have been developed and analyzed empirically. Most such algorithms have been established in the context of verifying the conjecture that trees are graceful.

In this paper, we present the first rigorous running time analysis of a simple evolutionary algorithm applied to finding labelings of graceful graphs. We prove that an evolutionary algorithm can find a graceful labeling in polynomial time for all paths, stars, and complete bipartite graphs with a constant-sized partition. We also empirically compare the running time of a simple evolutionary algorithm against a complete constraint solver.

References

[1]

Michal Adamaszek. 2006. Efficient enumeration of graceful permutations. CoRR math/0608513 (2006). http://arxiv.org/abs/math/0608513

[2]

R. E. L. Aldred and Brendan D. McKay. 1998. Graceful and harmonious labellings of trees. Bulletin of the Institute of Combinatorics and its Applications 23 (1998), 69--72.

[3]

Ibrahim Cahit Arkut, Refik Cahit Arkut, and Nasir Ghani. 2000. Graceful label numbering in optical MPLS networks. In OptiComm 2000: Optical Networking and Communications, Imrich Chlamtac (Ed.), Vol. 4233. International Society for Optics and Photonics, SPIE, 1 -- 8.

[4]

Jacqueline Ayel and Odile Favaron. 1981. The helms are graceful. Université Paris-Sud, Centre d'Orsay, Laboratoire de recherche en Informatique.

[5]

Wolfgang Banzhaf. 1994. Genotype-Phenotype-Mapping and Neutral Variation - A Case Study in Genetic Programming. In Parallel Problem Solving from Nature - PPSN III, International Conference on Evolutionary Computation. The Third Conference on Parallel Problem Solving from Nature, Jerusalem, Israel, October 9-14, 1994, Proceedings (Lecture Notes in Computer Science, Vol. 866), Yuval Davidor, Hans-Paul Schwefel, and Reinhard Männer (Eds.). Springer, 322--332.

[6]

Jeane-Claude Bermond. 1979. Graceful graphs, radio antennae and French windmills. In Proceedings of the One Day Combinatorics Conference (Research Notes in Mathematics, Vol. 34). Pitman, 18--37.

[7]

Gary S. Bloom and Solomon W. Golomb. 1977. Applications of Numbered Undirected Graphs. Proc. IEEE 65, 4 (April 1977), 562--570.

[8]

Ljiljana Brankovic and Ian M. Wanless. 2011. Graceful Labelling: State of the Art, Applications and Future Directions. Math. Comput. Sci. 5, 1 (2011), 11--20.

[9]

Benjamin Doerr, Daniel Johannsen, and Carola Winzen. 2012. Multiplicative Drift Analysis. Algorithmica 64 (2012), 673--697.

Digital Library

[10]

Jon Ernstberger and Andy D. Perkins. 2013. A Metaheuristic Search Technique for Grace fill Labels of Graphs. Technical Report. LaGrange College. Presented at the 44th annual Southeastern International Conference on Combinatorics, Graph Theory, and Computing.

[11]

Kourosh Eshghi and Parham Azimi. 2004. Applications of mathematical programming in graceful labeling of graphs. Journal of Applied Mathematics (2004), 1--8.

[12]

Wenjie Fang. 2010. A Computational Approach to the Graceful Tree Conjecture. CoRR abs/1003.3045 (2010). arXiv:1003.3045 http://arxiv.org/abs/1003.3045

[13]

Roberto Frucht. 1979. Graceful Numbering of Wheels and Related Graphs. Annals of the New York Academy of Sciences 319, 1 (1979), 219--229.

[14]

Joseph A. Gallian. 2021. A dynamic survey of graph labeling. Electronic Journal of Combinatorics (2021). Dynamic Surveys #DS6.

[15]

Gecode Team. 2019. Gecode: Generic Constraint Development Environment. Available from http://www.gecode.org.

[16]

Solomon W. Golomb. 1972. How To Number A Graph. In Graph Theory and Computing, Ronald C. Read (Ed.). Academic Press, 23--37.

[17]

Jun He and Xin Yao. 2004. A study of drift analysis for estimating computation time of evolutionary algorithms. Natural Computing 3 (2004), 21--35.

Digital Library

[18]

Michael Horton. 2003. Graceful Trees: Statistics and Algorithms. Bachelor's Thesis. University of Tasmania.

[19]

Clarissa Van Hoyweghen, Bart Naudts, and David E. Goldberg. 2002. Spin-Flip Symmetry and Synchronization. Evol. Comput. 10, 4 (2002), 317--344.

Digital Library

[20]

Ronan Le Bras, Carla P Gomes, and Bart Selman. 2013. Double-wheel graphs are graceful. In Twenty-Third International Joint Conference on Artificial Intelligence.

[21]

Houra Mahmoudzadeh and Kourosh Eshghi. 2007. A Metaheuristic Approach to the Graceful Labeling Problem of Graphs. In 2007 IEEE Swarm Intelligence Symposium. 84--91.

Digital Library

[22]

Karen E. Petrie and Barbara M. Smith. 2003. Symmetry Breaking in Graceful Graphs. In Principles and Practice of Constraint Programming - CP 2003, 9th International Conference, CP 2003, Kinsale, Ireland, September 29 - October 3, 2003, Proceedings (Lecture Notes in Computer Science, Vol. 2833), Francesca Rossi (Ed.). Springer, 930--934.

Digital Library

[23]

Steven David Prestwich and Andrea Roli. 2005. Symmetry Breaking and Local Search Spaces. In Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, Second International Conference, CPAIOR 2005, Prague, Czech Republic, May 30 - June 1, 2005, Proceedings (Lecture Notes in Computer Science, Vol. 3524), Roman Barták and Michela Milano (Eds.). Springer, 273--287.

Digital Library

[24]

Timothy A. Redl. 2003. Graceful Graphs and Graceful Labelings: Two Mathematical Programming Formulations and Some Other New Results. Congressus Numerantium 164 (2003), 17--31.

[25]

Alexander Rosa. 1967. On certain valuations of the vertices of a graph. In Theory of Graphs, International Symposium. Gordon and Breach; Dunod, 349--355.

[26]

Jens Scharnow, Karsten Tinnefeid, and Ingo Wegener. 2004. The analysis of evolutionary algorithms on sorting and shortest paths problems. J. Math. Model. Algorithms 3, 4 (2004), 349--366.

[27]

Barbara M. Smith and Jean-François Puget. 2010. Constraint models for graceful graphs. Constraints 15, 1 (2010), 64--92.

Digital Library

[28]

Ole Tange. 2011. GNU Parallel - The Command-Line Power Tool. ;login: The USENIX Magazine 36, 1 (Feb 2011), 42--47.

[29]

D. Thierens, D.E. Goldberg, and A.G. Pereira. 1998. Domino convergence, drift, and the temporal-salience structure of problems. In 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98TH8360). 535--540.

[30]

Carsten Witt. 2018. Domino convergence: why one should hill-climb on linear functions. In Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2018, Kyoto, Japan, July 15-19, 2018, Hernán E. Aguirre and Keiki Takadama (Eds.). ACM, 1539--1546.

Digital Library

[31]

Robert Alan Wright, Bruce Richmond, Andrew Odlyzko, and Brendan D. McKay. 1986. Constant Time Generation of Free Trees. SIAM J. Comput. 15, 2 (1986), 540--548.

Digital Library

Cited By

Régin FDe Maria E(2024)Generative Constraint Programming Revisited2024 IEEE 36th International Conference on Tools with Artificial Intelligence (ICTAI)10.1109/ICTAI62512.2024.00012(18-26)Online publication date: 28-Oct-2024
https://doi.org/10.1109/ICTAI62512.2024.00012
Branson LSutton AYan X(2023)Finding Antimagic Labelings of Trees by Evolutionary SearchProceedings of the 17th ACM/SIGEVO Conference on Foundations of Genetic Algorithms10.1145/3594805.3607133(27-37)Online publication date: 30-Aug-2023
https://dl.acm.org/doi/10.1145/3594805.3607133

Index Terms

Evolving labelings of graceful graphs
1. Theory of computation
  1. Theory and algorithms for application domains
    1. Theory of randomized search heuristics

Recommendations

Antimagic vertex labelings of hypergraphs

Hartsfield and Ringel (J. Recreat. Math. 21 (2) (1989) 107) conjectured that for simple, finite and connected graphs different from K₂ there exists an antimagic edge labeling; by dualization this yields a conjecture on vertex labelings of a special ...
Radio labelings of distance graphs

Motivated by the Channel Assignment Problem, we study radio k-labelings of graphs. A radio k-labeling of a connected graph G is an assignment c of non-negative integers to the vertices of G such that |c(x)-c(y)|>=k+1-d(x,y), for any two vertices x and y,...
Decompositions of complete graphs and complete bipartite graphs into isomorphic supersubdivision graphs

A graph H is called a supersubdivison of a graph G if H is obtained from G by replacing every edge uv of G by a complete bipartite graph K_2,m (m may vary for each edge) by identifying u and v with the two vertices in K_2,m that form one of the two ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

GECCO '22: Proceedings of the Genetic and Evolutionary Computation Conference

July 2022

1472 pages

ISBN:9781450392372

DOI:10.1145/3512290

Editor:
Jonathan E. Fieldsend
University of Exeter
,
General Chair:
Markus Wagner
The University of Adelaide

Copyright © 2022 ACM.

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 the author(s) 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

SIGEVO: ACM Special Interest Group on Genetic and Evolutionary Computation

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 08 July 2022

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

GECCO '22

Sponsor:

SIGEVO

GECCO '22: Genetic and Evolutionary Computation Conference

July 9 - 13, 2022

Massachusetts, Boston

Acceptance Rates

Overall Acceptance Rate 1,669 of 4,410 submissions, 38%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
78
Total Downloads

Downloads (Last 12 months)19
Downloads (Last 6 weeks)0

Reflects downloads up to 25 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Régin FDe Maria E(2024)Generative Constraint Programming Revisited2024 IEEE 36th International Conference on Tools with Artificial Intelligence (ICTAI)10.1109/ICTAI62512.2024.00012(18-26)Online publication date: 28-Oct-2024
https://doi.org/10.1109/ICTAI62512.2024.00012
Branson LSutton AYan X(2023)Finding Antimagic Labelings of Trees by Evolutionary SearchProceedings of the 17th ACM/SIGEVO Conference on Foundations of Genetic Algorithms10.1145/3594805.3607133(27-37)Online publication date: 30-Aug-2023
https://dl.acm.org/doi/10.1145/3594805.3607133

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Table of Conten