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A heuristic algorithm using tree decompositions for the maximum happy vertices problem

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Abstract

We propose a new methodology to develop heuristic algorithms using tree decompositions. Traditionally, such algorithms construct an optimal solution of the given problem instance through a dynamic programming approach. We modify this procedure by introducing a parameter W that dictates the number of dynamic programming states to consider. We drop the exactness guarantee in favour of a shorter running time. However, if W is large enough such that all valid states are considered, our heuristic algorithm proves optimality of the constructed solution. In particular, we implement a heuristic algorithm for the Maximum Happy Vertices problem using this approach. Our algorithm more efficiently constructs optimal solutions compared to the exact algorithm for graphs of bounded treewidth. Furthermore, our algorithm constructs higher quality solutions than state-of-the-art heuristic algorithms Greedy-MHV and Growth-MHV for instances of which at least 40% of the vertices are initially coloured, at the cost of a larger running time.

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Availability of data and materials

The instances used for experimentation are available https://github.com/LouisCarpentier42/HeuristicAlgorithmsUsingTreeDecompositions/tree/main/instances. The results of all experiments are available on https://github.com/LouisCarpentier42/HeuristicAlgorithmsUsingTreeDecompositions/tree/main/results.

Code Availability

All code is publicly available on https://github.com/LouisCarpentier42/HeuristicAlgorithmsUsingTreeDecompositions.

Notes

  1. https://pacechallenge.org/2017/.

  2. http://users.cecs.anu.edu.au/~bdm/data/graphs.html.

  3. We thank Prof. Marco Ghirardi for providing us the instances.

  4. We thank Prof. Marco Ghirardi for providing us with the source code of the matheuristic.

References

  • Agrawal, A.: On the parameterized complexity of happy vertex coloring. In: International Workshop on Combinatorial Algorithms, Springer, pp. 103–115 (2017)

  • Agrawal, A., Aravind, N., Kalyanasundaram, S., et al.: Parameterized complexity of happy coloring problems. Theoret. Comput. Sci. 835, 58–81 (2020)

    Article  MathSciNet  Google Scholar 

  • Aravind, N., Kalyanasundaram, S., Kare, A.S.: Linear time algorithms for happy vertex coloring problems for trees. In: International Workshop on Combinatorial Algorithms, Springer, pp. 281–292 (2016)

  • Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Algebraic Disc. Methods 8(2), 277–284 (1987)

    Article  MathSciNet  Google Scholar 

  • Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM (JACM) 41(1), 153–180 (1994)

    Article  MathSciNet  Google Scholar 

  • Bannach, M., Berndt, S.: Practical access to dynamic programming on tree decompositions. Algorithms 12(8), 172 (2019)

    Article  MathSciNet  Google Scholar 

  • Blum, C., Pinacho, P., López-Ibáñez, M., et al.: Construct, merge, solve & adapt a new general algorithm for combinatorial optimization. Comput. Oper. Res. 68, 75–88 (2016)

    Article  MathSciNet  Google Scholar 

  • Bodlaender, H.L., Bonsma, P., Lokshtanov, D.: The fine details of fast dynamic programming over tree decompositions. In: International Symposium on Parameterized and Exact Computation, Springer, pp. 41–53 (2013)

  • Bodlaender, H.L., Drange, P.G., Dregi, M.S., et al.: A c\(^\text{ k }n\) 5-approximation algorithm for treewidth. SIAM J. Comput. 45(2), 317–378 (2016)

    Article  MathSciNet  Google Scholar 

  • Carpentier, L.: Developing heuristic algorithms for graph optimisation problems using tree decompositions. Master’s thesis (Supervisors: Prof. Jan Goedgebeur and Jorik Jooken), KU Leuven. Faculteit Ingenieurswetenschappen (2022)

  • Coolsaet, K., D’hondt, S., Goedgebeur, J.: House of graphs 2.0: A database of interesting graphs and more. Discrete Appl. Math. 325, 97–107 (2023)

  • Cygan, M., Fomin, F.V., Kowalik, Ł, et al.: Parameterized Algorithms, vol. 5. Springer, Cham, Switzerland (2015)

    Book  Google Scholar 

  • Dell, H., Komusiewicz, C., Talmon, N., et al.: The PACE 2017 parameterized algorithms and computational experiments challenge: The second iteration. In: 12th International Symposium on Parameterized and Exact Computation (IPEC 2017), Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2018)

  • Easley, D., Kleinberg, J.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, New York, NY (2010)

    Book  Google Scholar 

  • Erdős, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5(1), 17–60 (1960)

    MathSciNet  Google Scholar 

  • Gao, H., Gao, W.: Kernelization for maximum happy vertices problem. In: Latin American Symposium on Theoretical Informatics, Springer, pp. 504–514 (2018)

  • Ghirardi, M., Salassa, F.: A simple and effective algorithm for the maximum happy vertices problem. TOP 30(1), 181–193 (2022)

    Article  MathSciNet  Google Scholar 

  • Hamann, M., Strasser, B.: Graph bisection with pareto optimization. J. Exp. Algorithmics (JEA) 23, 1–34 (2018)

    MathSciNet  Google Scholar 

  • Hutter, F., Hoos, H.H., Leyton-Brown, K.: Sequential model-based optimization for general algorithm configuration. In: International conference on learning and intelligent optimization, Springer, pp. 507–523 (2011)

  • Jooken, J., Leyman, P., De Causmaecker, P.: A multi-start local search algorithm for the hamiltonian completion problem on undirected graphs. Journal of Heuristics 26(5), 743–769 (2020)

    Article  Google Scholar 

  • Karp, R.M.: Reducibility Among Combinatorial Problems. Springer (2010)

  • Kőnig, D.: Gráfok és mátrixok. Matematikai és Fizikai Lapok 38, 116–119 (1931)

    Google Scholar 

  • Lewis, R., Thiruvady, D., Morgan, K.: Finding happiness: an analysis of the maximum happy vertices problem. Comput. Oper. Res. 103, 265–276 (2019)

    Article  MathSciNet  Google Scholar 

  • Minty, G.J.: On maximal independent sets of vertices in claw-free graphs. J. Comb. Theory, Series B 28(3), 284–304 (1980)

    Article  MathSciNet  Google Scholar 

  • Peeters, F.: Oplossingsmethodes voor het maximum happy vertices probleem. Master’s thesis (Supervisor: Prof. Patrick De Causmaecker), KU Leuven. Faculteit Ingenieurswetenschappen (2020)

  • Robertson, N., Seymour, P.D.: Graph minors. III. Planar tree-width. J. Comb. Theory, Ser. B 36(1), 49–64 (1984)

    Article  MathSciNet  Google Scholar 

  • Tamaki, H.: Positive-instance driven dynamic programming for treewidth. J. Comb. Optim. 37(4), 1283–1311 (2019)

    Article  MathSciNet  Google Scholar 

  • Thiruvady, D., Lewis, R., Morgan, K.: Tackling the maximum happy vertices problem in large networks. 4OR 18, 507–527 (2020)

    Article  MathSciNet  Google Scholar 

  • Zhang, P., Li, A.: Algorithmic aspects of homophyly of networks. Theoret. Comput. Sci. 593, 117–131 (2015)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

The research of Jan Goedgebeur was supported by Internal Funds of KU Leuven. Jorik Jooken is supported by a Postdocoral Fellowship of the Research Foundation Flanders (FWO) with contract number 1222524N. We gratefully acknowledge the support provided by the ORDinL project (FWO-SBO S007318N, Data Driven Logistics, 1/1/2018 - 31/12/2021). This research also received funding from the Flemish Government under the “Onderzoeksprogramma Artificiële Intelligentie (AI) Vlaanderen” programme. The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government - department EWI.

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Authors and Affiliations

  1. Department of Computer Science, KU Leuven Brugge, 8200, Bruges, Belgium

    Louis Carpentier

  2. Department of Computer Science, KU Leuven KULAK, 8500, Kortrijk, Belgium

    Jorik Jooken & Jan Goedgebeur

  3. Department of Applied Mathematics, Computer Science and Statistics, Ghent University, 9000, Ghent, Belgium

    Jan Goedgebeur

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  1. Louis Carpentier
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  2. Jorik Jooken
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  3. Jan Goedgebeur
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Correspondence to Louis Carpentier.

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Carpentier, L., Jooken, J. & Goedgebeur, J. A heuristic algorithm using tree decompositions for the maximum happy vertices problem. J Heuristics 30, 67–107 (2024). https://doi.org/10.1007/s10732-023-09522-x

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