Skip to main content
SpringerLink
Log in

Research Horizons

  • Chapter
Fundamentals of Parameterized Complexity

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

  • 3365 Accesses

Abstract

In this final chapter, we describe what we think are some of the most important open problems of the field. In the first book, Downey and Fellows (Parameterized Complexity. Monographs in Computer Science, Springer, Berlin, 1999), we offered two lists of problems, 18 altogether, that we then thought significant and especially challenging. As this book goes to press, 12 of these have been resolved! Many of the solutions to these problems involved significant new ideas and advances in the field.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Ajtai, The shortest vector problem in ℓ 2 is NP-hard for randomized reductions, in Proceedings of 30th ACM Symposium on Theory of Computing (STOC ’98), Dallas, TX, USA, May 23–26, 1998, ed. by J. Vitter (ACM, New York, 1998), pp. 10–19

    Google Scholar 

  2. H. Björklund, S. Sandberg, S. Vorobyov, On fixed-parameter complexity of infinite games, in The Nordic Workshop on Programming Theory (NWPT’03), Åbo Akademi University, Turku, Finland, October 29–31, 2003, ed. by K. Sere, M. Waldén, A. Karlsson (Åbo Akademi, Department of Computer Science, Turku, 2003), pp. 62–64

    Google Scholar 

  3. J. Blasiok, M. Kaminski, Chain minors are FPT, in Proceedings of IPEC 2013 (2013)

    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. Cygan, D. Marx, M. Pilipczuk, M. Pilipczuk, The planar directed k-Vertex-Disjoint Paths problem is fixed-parameter tractable, arXiv:1304.4207

  6. E. Demaine, M. Hajiaghayi, K.-i. Kawarabayashi, Contraction decomposition in H-minor-free graphs and algorithmic applications, in Proceedings of 43rd ACM Symposium on Theory of Computing (STOC ’11), San Jose, CA, USA, June 6–June 8, 2011, ed. by L. Fortnow, S. Vadhan (ACM, New York, 2011), pp. 441–450

    Google Scholar 

  7. R. Downey, M. Fellows, Fixed-parameter tractability and completeness II: On completeness for W[1]. Theor. Comput. Sci. 141(1–2), 109–131 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  9. R. Downey, M. Fellows, A. Vardy, G. Whittle, The parametrized complexity of some fundamental problems in coding theory. SIAM J. Comput. 29(2), 545–570 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. T. Eiter, G. Gottlob, Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput. 24, 1278–1304 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Elbassioni, M. Hagen, I. Rauf, Some fixed-parameter tractable classes of hypergraph duality and related problems, in Parameterized and Exact Computation, 3rd International Workshop, IWPEC ’08, Revised Selected Papers, Victoria, Canada, May 14–16, 2008, ed. by M. Grohe, R. Niedermeier. LNCS, vol. 5018 (Springer, Berlin, 2008), pp. 91–102

    Chapter  Google Scholar 

  12. M. Fellows, N. Koblitz, Fixed-parameter complexity and cryptography, in Proceedings of the 10th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 10), Puerto Rico, May 1993, ed. by G. Cohen, T. Mora, O. Moreno. LNCS, vol. 673 (Springer, Berlin, 1993), pp. 121–131

    Chapter  Google Scholar 

  13. F. Fomin, P. Golovach, D. Thilikos, Contraction bidimensionality: the accurate picture, in Algorithms—ESA 2009: Proceedings of 17th Annual European Symposium, Copenhagen, Denmark, September 7–9, 2009, ed. by A. Fiat, P. Sanders. LNCS, vol. 5757 (Springer, Berlin, 2009), pp. 706–717

    Chapter  Google Scholar 

  14. F. Fomin, D. Lokshtanov, V. Raman, S. Saurabh, Fast local search algorithm for weighted feedback arc set in tournaments, in Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2010, Atlanta, Georgia, USA, July 11–15, 2010 (AAAI Press, Menlo Park, 2010)

    Google Scholar 

  15. F. Fomina, D. Marx, FPT suspects and tough customers: open problems of Downey and Fellows, in The Multivariate Algorithmic Revolution and Beyond: Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, ed. by H. Bodlaender, R. Downey, F. Fomin, D. Marx. LNCS, vol. 7370 (Springer, Berlin, 2012), pp. 457–468

    Chapter  Google Scholar 

  16. J. Geelen, B. Gerards, G. Whittle, Towards a structure theory for matrices and matroids, in International Congress of Mathematicians, Madrid, August 22–30, 2006, vol. III, ed. by M. Sanz-Solé, J. Soria, J. Varona, J. Verdera (Eur. Math. Soc., Zürich, 2006), pp. 827–842

    Google Scholar 

  17. J. Geelen, B. Gerards, G. Whittle, Rota’s conjecture, to appear. Result announced at the British Combinatorial Conference, 2013

    Google Scholar 

  18. I. Giotis, V. Guruswami, Correlation clustering with a fixed number of clusters. Theory Comput. 2, 249–266 (2006)

    Article  MathSciNet  Google Scholar 

  19. S. Guillemot, D. Marx, Finding small patterns in permutations in linear time, arXiv:1307.3073

  20. J. Gustedt, Well quasi ordering finite posets and formal languages. J. Comb. Theory, Ser. B 65(1), 111–124 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Hartung, R. Niedermeier, Incremental list coloring of graphs, parameterized by conservation. Theor. Comput. Sci. 494, 86–98 (2013)

    Article  MathSciNet  Google Scholar 

  22. R. Hemmecke, S. Onn, L. Romanchuk, N-fold integer programming in cubic time (hence multiway tables are fixed-parameter tractable) (with Raymond Hemmecke and Lyubov Romanchuk). Math. Program. 137, 325–341 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Hliněný, G. Whittle, Matroid tree-width. Eur. J. Comb. 27, 1117–1128 (2006)

    Article  MATH  Google Scholar 

  24. E. Luks, Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. Syst. Sci. 25, 42–65 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  25. D. Marx, Parameterized complexity and approximation algorithms. Comput. J. 51(1), 60–78 (2008)

    Article  Google Scholar 

  26. D. Marx, Searching the k-change neighborhood for TSP is W[1]-hard. Oper. Res. Lett. 36(1), 31–36 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Oum, P. Seymour, Approximating clique-width and branch-width. J. Comb. Theory, Ser. B 96, 514–528 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Scott, On the parameterized complexity of finding short winning strategies in combinatorial games, PhD thesis, University of Victoria, 2009

    Google Scholar 

  29. S. Szeider, The parameterized complexity of k-flip local search for sat and max sat. Discrete Optim. 8(1), 139–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Vardy, Algorithmic complexity in coding theory and the minimum distance problem, in Proceedings of 29th ACM Symposium on Theory of Computing (STOC ’97), Baltimore, MD, USA, May 4–6, 1997, ed. by F. Leighton, P. Shor (ACM, New York, 1997), pp. 92–109

    Google Scholar 

  31. D. Vertigan, G. Whittle, Recognising polymatroids associated with hypergraphs. Comb. Probab. Comput. 2, 519–530 (1993)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

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

Authors
  1. Rodney G. Downey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael R. Fellows
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag London

About this chapter

Cite this chapter

Downey, R.G., Fellows, M.R. (2013). Research Horizons. In: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-5559-1_33

Download citation

  • DOI: https://doi.org/10.1007/978-1-4471-5559-1_33

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-5558-4

  • Online ISBN: 978-1-4471-5559-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation