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Iterative Compression, and Measure and Conquer, for Minimization Problems

  • Chapter
Fundamentals of Parameterized Complexity

Part of the book series: Texts in Computer Science ((TCS))

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Abstract

We introduce the technique of iterative compression. We illustrate how this can be combined with analytical techniques such as measure and conquer.

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Notes

  1. 1.

    Strictly speaking the kernel is smaller than this, namely 3k−c 1−ρ(V 1) where ρ(V 1) is the size of a maximum matching of the subgraph induced by the set \(V_{1}'\) of vertices of V 1 of degree larger than 3. However, the present bound is sufficient for our purposes and we refer the reader to the journal version of [130] for analysis.

  2. 2.

    Not, of course, to be confused with the notion of independent set in a graph.

  3. 3.

    It is not linear since α(n) is unbounded.

References

  1. H. Bodlaender, A cubic kernel for feedback vertex set, in Proceedings of 24th Annual Symposium on Theoretical Aspects of Computer Science, STACS 2007, Aachen, Germany, February 22–24, 2007, ed. by W. Thomas, P. Weil. LNCS, vol. 4393 (Springer, Berlin, 2007), pp. 320–331

    Google Scholar 

  2. Y. Cao, J. Chen, Y. Liu, On feedback vertex set: new measure and new structures, in Proceedings of the 12th Scandinavian Symposium and Workshops on Algorithm Theory, ed. by H. Kaplan (Springer, Berlin, 2010), pp. 93–104

    Google Scholar 

  3. J. Chen, F. Fomin, Y. Liu, S. Lu, Y. Villanger, Improved algorithms for feedback vertex set problems. J. Comput. Syst. Sci. 74(7), 1188–1198 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Chen, Y. Liu, S. Lu, B. O’Sullivan, I. Razgon, A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55(5), 1–19 (2008)

    MathSciNet  Google Scholar 

  5. M. Davis, H. Putnam, A computing procedure for quantification theory. J. ACM 7, 201–215 (1960)

    MathSciNet  MATH  Google Scholar 

  6. F. Dehne, M. Fellows, F. Rosamond, P. Shaw, Greedy localization, iterative compression, and modeled crown reductions: new FPT techniques, an improved algorithm for set splitting, and a novel 2k kernelization for vertex cover, in Parameterized and Exact Computation, Proceedings of First International Workshop, IWPEC 2004, Bergen, Norway, September 14–17, 2004, ed. by R. Downey, M. Fellows, F. Dehne. LNCS, vol. 3162 (Springer, Berlin, 2004), pp. 271–281

    Google Scholar 

  7. R. Downey, M. Fellows, Parameterized Complexity. Monographs in Computer Science (Springer, Berlin, 1999)

    Book  Google Scholar 

  8. F. Fomin, F. Grandoni, D. Kratsch, Measure and conquer: domination—a case study, in Proceedings of 32nd International Colloquium on Automata, Languages and Programming (ICALP 2005), Lisbon, Portugal, July 11–15, 2005, ed. by L. Caires, G. Italiano, L. Monteiro, C. Palamidessi, M. Yung. LNCS, vol. 3580 (Springer, Berlin, 2005), pp. 191–203

    Chapter  Google Scholar 

  9. F. Fomin, F. Grandoni, D. Kratsch, Measure and conquer: a simple O(20.288n) independent set algorithm, in Proceedings of the Seventeenth Annual ACM–SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22–26, 2006 (ACM, New York, 2006), pp. 18–25

    Google Scholar 

  10. F. Fomin, F. Grandoni, D. Kratsch, Solving connected dominating set faster than 2n, in FSTTCS 2006: Foundations of Software Technology and Theoretical Computer Science, Proceedings of 26th International Conference, Kolkata, India, December 13–15, 2006, ed. by S. Arun-Kumar, N. Garg. LNCS, vol. 4337 (Springer, Berlin, 2006), pp. 152–163

    Chapter  Google Scholar 

  11. F. Fomin, F. Grandoni, A. Pyatkin, A. Stepanov, Bounding the number of minimal dominating sets: a measure and conquer approach, in Algorithms and Computation: Proceedings of 16th International Symposium, ISAAC 2005, Sanya, Hainan, China, December 19–21, 2005, ed. by X. Deng, D. Du. LNCS, vol. 3827 (Springer, Berlin, 2005), pp. 573–582

    Chapter  Google Scholar 

  12. H. Gabow, M. Stallmann, An augmenting path algorithm for linear matroid parity. Combinatorica 6(2), 123–150 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. D.J. Guan, Generalized Gray codes with applications. Proc. Natl. Sci. Counc., Repub. China 22, 841–848 (1998)

    Google Scholar 

  14. J. Guo, J. Gramm, F. Hüffner, R. Niedermeier, S. Wernicke, Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. J. Comput. Syst. Sci. 72(8), 1386–1396 (2006)

    Article  MATH  Google Scholar 

  15. J. Guo, H. Moser, R. Niedermeier, Iterative compression for exactly solving NP-hard minimization problems, in Algorithmics of Large and Complex Networks, ed. by J. Lerner, D. Wagner, K. Zweig. LNCS, vol. 5515 (Springer, Berlin, 2009), pp. 65–80

    Chapter  Google Scholar 

  16. G. Gutin, E. Kim, M. Lampis, V. Mitsou, Vertex cover problem parameterized above and below tight bounds. Theory Comput. Syst. 48(2), 402–410 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Hüffner, Algorithm engineering for optimal graph bipartization, in WEA 2005: 4th International Workshop on Efficient and Experimental Algorithms, Santorini Island, Greece, May 10–13, 2005, ed. by S. Nikoletseas. LNCS, vol. 3503 (Springer, Berlin, 2005), pp. 240–252

    Google Scholar 

  18. F. Hüffner, C. Komusiewicz, H. Moser, R. Niedermeier, Fixed-parameter algorithms for cluster vertex deletion. Theory Comput. Syst. 47, 196–217 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Y. Iwata, A faster algorithm for dominating set analyzed by the potential method, in Parameterized and Exact Computation, 6th International Symposium, IPEC ’11, Revised Selected Papers, Saarbrücken, Germany, September 6–8, 2011, ed. by D. Marx, P. Rossmanith. LNCS, vol. 7112 (Springer, Berlin, 2011), pp. 41–54

    Chapter  Google Scholar 

  20. R. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations, ed. by R. Miller, J. Thatcher (Plenum, New York, 1972), pp. 45–68

    Google Scholar 

  21. D. Kratsch, M. Liedloff, An exact algorithm for the minimum dominating clique problem, in Parameterized and Exact Computation, Proceedings of Second International Workshop, IWPEC ’06, Zürich, Switzerland, September 13–15, 2006, ed. by H. Bodlaender, M. Langston. LNCS, vol. 4169 (Springer, Berlin, 2006), pp. 78–89

    Google Scholar 

  22. D. Lokshtanov, N.S. Narayanaswamy, V. Raman, M.S. Ramanujan, S. Saurabh, Faster parameterized algorithms using linear programming, arXiv:1203.0833 [cs.DS] (2012)

  23. R. Niedermeier, P. Rossmanith, A general method to speed up fixed-parameter-tractable algorithms. Inf. Process. Lett. 73(3–4), 125–129 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. I. Razgon, B. O’Sullivan, Almost 2-SAT is fixed-parameter tractable. J. Comput. Syst. Sci. 75(8), 435–450 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. B. Reed, K. Smith, A. Vetta, Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. J. van Rooij, H. Bodlaender, Design by measure and conquer: a faster algorithm for dominating set, in Proceedings of 25th Annual Symposium on Theoretical Aspects on Computer Science, STACS 2008, Bordeau, France, February 21–23, 2008, ed. by S. Albers, P. Weil. Leibniz International Proceedings in Informatics, vol. 1 (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, 2008), pp. 657–668

    Google Scholar 

  27. S. Wernicke, On the algorithmic tractability of single nucleotide polymorphism (SNP) analysis and related problems, PhD thesis, Wilhelm-Schickard-Institute für Informatik, Universität Tübingen, 2003

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Statistics and Operations Research, Victoria University, Wellington, New Zealand

    Rodney G. Downey

  2. School of Engineering and Information Technology, Charles Darwin University, Darwin, Northern Territory, Australia

    Michael R. Fellows

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  1. Rodney G. Downey
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Downey, R.G., Fellows, M.R. (2013). Iterative Compression, and Measure and Conquer, for Minimization Problems. In: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-5559-1_6

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  • DOI: https://doi.org/10.1007/978-1-4471-5559-1_6

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-5558-4

  • Online ISBN: 978-1-4471-5559-1

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