Optimization Letters

, Volume 10, Issue 1, pp 33–45 | Cite as

Approximating the minimum hub cover problem on planar graphs

  • Belma YelbayEmail author
  • Ş. İlker Birbil
  • Kerem Bülbül
  • Hasan Jamil
Original Paper


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.


Approximation algorithm Query processing Subgraph isomorphism Planar graph decomposition Minimum hub cover problem 

Mathematics Subject Classification

68T20 90C27 05C90 


Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Baker, B.: Approximation algorithms for \({\cal NP}\)-complete problems. J. Assoc. Comput. Mach. 41, 153–180 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bienstock, D., Monma, C.: On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica 5, 93–109 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cook, S.: The complexity of theorem-proving procedures. In: 3rd ACM Symposium on Theory of Computing, pp. 151–158. Ohio (1971)Google Scholar
  5. 5.
    Garey, M., Johnson, D., Stockmeyer, L.: Some simplified np-complete graph problems. Theor. Comput. Sci. 1, 237–267 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Hopcroft, J., Tarjan, R.: Efficient planarity testing. J. ACM 21, 549–568 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Jamil, H.M.: Computing subgraph isomorphic queries using structural unification and minimum graph structures. In: SAC, pp. 1053–1058 (2011)Google Scholar
  8. 8.
    Kammer, F.: Determining the smallest \(k\) such that \(g\) is \(k-\)outerplanar. Lect. Notes Comput. Sci. 4698, 359–370 (2007)CrossRefGoogle Scholar
  9. 9.
    Lipets, V., Vanetik, N., Gudes, E.: Subsea: an efficient heuristic algorithm for subgraph isomorphism. Data Min. Knowl. Discov. 19, 320–350 (2009)MathSciNetCrossRefGoogle Scholar
  10. 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)CrossRefGoogle Scholar
  11. 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)Google Scholar
  12. 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)Google Scholar
  13. 13.
    Rivero, C.R., Jamil, H.M.: Exact subgraph isomorphism using graphlets and minimum hub covers (2014) (Work-in-process)Google Scholar
  14. 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)Google Scholar
  15. 15.
    Ullmann, J.: An algorithm for subgraph isomorphism. J. ACM 23, 31–42 (1976)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Weber, M., Liwicki, M., Dengel, A.: Faster subgraph isomorphism detection by well-founded total order indexing. Pattern Recognit. Lett. 33, 2011–2019 (2012)CrossRefGoogle Scholar
  17. 17.
    Yelbay, B.: Minimum hub cover problem: Solution methods and applications. Ph.D. thesis, Sabanci University (2014)Google Scholar
  18. 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. 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)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Belma Yelbay
    • 1
    Email author
  • Ş. İlker Birbil
    • 1
  • Kerem Bülbül
    • 1
  • Hasan Jamil
    • 2
  1. 1.Manufacturing Systems and Industrial EngineeringSabancı UniversityIstanbulTurkey
  2. 2.Department of Computer ScienceUniversity of IdahoMoscowUSA

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