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Theory of Computing Systems

, Volume 60, Issue 1, pp 3–19 | Cite as

Incremental Problems in the Parameterized Complexity Setting

  • Bernard MansEmail author
  • Luke Mathieson
Article
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Abstract

Dynamic systems are becoming steadily more important with the profusion of mobile and distributed computing devices. Coincidentally incremental computation is a natural approach to deal with ongoing changes. We explore incremental computation in the parameterized complexity setting and show that incrementalization leads to non-trivial complexity classifications. Interestingly, some incremental versions of hard problems become tractable, while others remain hard. Moreover tractability or intractability is not a simple function of the problem’s static complexity, every level of the W-hierarchy exhibits complete problems with both tractable and intractable incrementalizations. For problems that are already tractable in their static form, we also show that incrementalization can lead to interesting algorithms, improving upon the trivial approach of using the static algorithm at each step.

Keywords

Algorithm complexity Incremental computation Kernelization Parameterized complexity Tractability 

Notes

Acknowledgments

This research was supported by Australian Research Council Grants DP110104560 and DP140100118.

References

  1. 1.
    Batagelj, V., Zaversnik, M.: AnO(m)algorithm for cores decomposition of networks. arXiv: cs/0310049 (2003)
  2. 2.
    Crowston, R., Gutin, G., Jones, M., Raman, V., Saurabh, S.: Parameterized Complexity of MaxSat Above Average. In: Theoretical Informatics - 10th Latin American Symposium LATIN’12. LNCS 7256, pp. 184–194 (2012)Google Scholar
  3. 3.
    Desikan, P., Pathak, N., Srivastava, J., Kumar, V.: Incremental page rank computation on evolving graphs. In: Special interest tracks and posters of the 14th international conference on World Wide Web, WWW’05, pp. 1094–1095. ACM (2005)Google Scholar
  4. 4.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer (1999)Google Scholar
  5. 5.
    Fernau, H., Schmid, M.L., Villanger, Y.: On the parameterised complexity of string morphism problems. Theory of Computing Systems 59(1), 24–51 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Flum, J., Grohe, M.: Parameterized complexity theory. Springer (2006)Google Scholar
  7. 7.
    Guo, J., Niedermeier, R., Wernicke, S.: Parameterized complexity of vertex cover variants. Theory of Computing Systems 41(3), 501–520 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Held, M., Huber, S.: Topology-oriented incremental computation of voronoi diagrams of circular arcs and straight-line segments. Comput. Aided Des. 41(5), 327–338 (2009)CrossRefzbMATHGoogle Scholar
  9. 9.
    Jakub Ł., Sankowski, P.: Optimal decremental connectivity in planar graphs. Theory of Computing Systems 59(1), 1–17 (2016)Google Scholar
  10. 10.
    Marx, D.: The Parameterized complexity approximation algorithms. Comput. J. 51(1), 60–78 (2008)CrossRefGoogle Scholar
  11. 11.
    Mathieson, L.: The parameterized complexity of editing graphs for bounded degeneracy. Theor. Comput. Sci. 411(34-36), 3181–3187 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Miltersen, P.B., Subramanian, S., Vitter, J.S., Tamassia, R.: Complexity models for incremental computation. Theor. Comput. Sci. 130(1), 203–236 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mouawad, A.E., Nishimura, N., Raman, V., Simjour, N., Suzuki, A.: On the Parameterized Complexity of Reconfiguration Problems. In: Proceedings of the 8th International Symposium on Parameterized and Exact Computation, IPEC’13, pp. 281–294. Springer (2013)Google Scholar
  14. 14.
    Protti, F., da Silva, M.D., Szwarcfiter, J.L.: Applying Modular Decomposition to Parameterized Cluster Editing Problems. Theory of Computing Systems 44(1), 91–104 (2009)Google Scholar
  15. 15.
    Ramalingam, G., Reps, T.: A categorized bibliography on incremental computation. In: Proceedings of the 20th ACM SIGPLAN-SIGACT symposium on Principles of programming languages POPL’93, pp. 502–510. ACM (1993)Google Scholar
  16. 16.
    Ramalingam, G., Reps, T.: On the computational complexity of dynamic graph problems. Theoretical Computer Science 158(1–2), 233–277 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sundaresh, R.S., Hudak, P.: Incremental computation via partial evaluation. In: Proceedings of the 18th ACM SIGPLAN-SIGACT symposium on Principles of programming languages POPL’91, pp. 1–13. ACM (1991)Google Scholar
  18. 18.
    Weber, V., Schwentick, T.: Dynamic complexity theory revisited. Theory of Computing Systems 40(4), 355–377 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of ComputingMacquarie UniversitySydneyAustralia

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