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Approximating the minimum hub cover problem on planar graphs

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Abstract

We study an approximation algorithm with a performance guarantee to solve a new \(\mathcal {NP}\)-hard optimization problem on planar graphs. The problem, which is referred to as the minimum hub cover problem, has recently been introduced to the literature to improve query processing over large graph databases. Planar graphs also arise in various graph query processing applications, such as; biometric identification, image classification, object recognition, and so on. Our algorithm is based on a well-known graph decomposition technique that partitions the graph into a set of outerplanar graphs and provides an approximate solution with a proven performance ratio. We conduct a comprehensive computational experiment to investigate the empirical performance of the algorithm. Computational results demonstrate that the empirical performance of the algorithm surpasses its guaranteed performance. We also apply the same decomposition approach to develop a decomposition-based heuristic, which is much more efficient than the approximation algorithm in terms of computation time. Computational results also indicate that the efficacy of the decomposition-based heuristic in terms of solution quality is comparable to that of the approximation algorithm.

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References

  1. Baker, B.: Approximation algorithms for \({\cal NP}\)-complete problems. J. Assoc. Comput. Mach. 41, 153–180 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baloch, S., Krim, H.: Object recognition through topo-geometric shape models using error-tolerant subgraph isomorphisms. IEEE Trans. Image Process. 19, 1191–1200 (2010)

    Article  MathSciNet  Google Scholar 

  3. Bienstock, D., Monma, C.: On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica 5, 93–109 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cook, S.: The complexity of theorem-proving procedures. In: 3rd ACM Symposium on Theory of Computing, pp. 151–158. Ohio (1971)

  5. Garey, M., Johnson, D., Stockmeyer, L.: Some simplified np-complete graph problems. Theor. Comput. Sci. 1, 237–267 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hopcroft, J., Tarjan, R.: Efficient planarity testing. J. ACM 21, 549–568 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  7. Jamil, H.M.: Computing subgraph isomorphic queries using structural unification and minimum graph structures. In: SAC, pp. 1053–1058 (2011)

  8. Kammer, F.: Determining the smallest \(k\) such that \(g\) is \(k-\)outerplanar. Lect. Notes Comput. Sci. 4698, 359–370 (2007)

    Article  Google Scholar 

  9. Lipets, V., Vanetik, N., Gudes, E.: Subsea: an efficient heuristic algorithm for subgraph isomorphism. Data Min. Knowl. Discov. 19, 320–350 (2009)

    Article  MathSciNet  Google Scholar 

  10. Llados, J., Marti, E., Villanueva, J.: Symbol recognition by error-tolerant subgraph matching between region adjacency graphs. IEEE Trans. Pattern Anal. Mach. Intell. 23, 1137–1143 (2001)

    Article  Google Scholar 

  11. Neuhaus, M., Bunke, H.: A graph matching based approach to fingerprint classification using directional variance. In: Audio-and Video-Based Biometric Person Authentication, Lecture Notes in Computer Science, vol. 3546, pp. 191–200 (2004)

  12. Rivero, C., Jamil, H.M.: On isomorphic matching of large disk resident graphs using an xquery engine. International Workshop on Graph Data Management, Techniques and Applications (2014)

  13. Rivero, C.R., Jamil, H.M.: Exact subgraph isomorphism using graphlets and minimum hub covers (2014) (Work-in-process)

  14. Shang, H., Zhang, Y., Lin, X., Yu, J.: Taming verification hardness: An efficient algorithm for testing subgraph isomorphism. In: Journal Proceedings of the VLDB Endowment, vol. 1, pp. 364–375. Auckland, New Zealand (2008)

  15. Ullmann, J.: An algorithm for subgraph isomorphism. J. ACM 23, 31–42 (1976)

    Article  MathSciNet  Google Scholar 

  16. Weber, M., Liwicki, M., Dengel, A.: Faster subgraph isomorphism detection by well-founded total order indexing. Pattern Recognit. Lett. 33, 2011–2019 (2012)

    Article  Google Scholar 

  17. Yelbay, B.: Minimum hub cover problem: Solution methods and applications. Ph.D. thesis, Sabanci University (2014)

  18. Yelbay, B., Ş. İ. Birbil, Bülbül, K., Jamil, H.M.: Trade-offs computing minimum hub cover toward optimized graph query processing (2013). arXiv:1311.1626

  19. Zhu, K., Zhang, Y., Lin, X., Zhu, G., Wang, W.: A novel and efficient framework for finding subgraph isomorphism mappings in large graphs. In: 15th International Conference on Database Systems for Advanced Applications, pp. 140–154. Tsukuba, Japan (2010)

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The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. Manufacturing Systems and Industrial Engineering, Sabancı University, Orhanlı-Tuzla, 34956, Istanbul, Turkey

    Belma Yelbay, Ş. İlker Birbil & Kerem Bülbül

  2. Department of Computer Science, University of Idaho, Moscow, ID, USA

    Hasan Jamil

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  1. Belma Yelbay
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  2. Ş. İlker Birbil
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  3. Kerem Bülbül
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  4. Hasan Jamil
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Correspondence to Belma Yelbay.

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Yelbay, B., Birbil, Ş.İ., Bülbül, K. et al. Approximating the minimum hub cover problem on planar graphs. Optim Lett 10, 33–45 (2016). https://doi.org/10.1007/s11590-015-0876-5

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  • DOI: https://doi.org/10.1007/s11590-015-0876-5

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