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What’s Next? Future Directions in Parameterized Complexity

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The Multivariate Algorithmic Revolution and Beyond

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7370))

Abstract

The progress in parameterized complexity has been very significant in recent years, with new research questions and directions, such as kernelization lower bounds, appearing and receiving considerable attention. This speculative article tries to identify new directions that might become similar hot topics in the future. First, we point out that the search for optimality in parameterized complexity already has good foundations, but lots of interesting work can be still done in this area. The systematic study of kernelization became a very successful research direction in recent years. We look at what general conclusions one can draw from these results and we argue that the systematic study of other algorithmic techniques should be modeled after the study of kernelization. In particular, we set up a framework for understanding which problems can be solved by branching algorithms. Finally, we discuss that the domain of directed graph problems is a challenging area which can potentially see significant progress in the following years.

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References

  1. Abu-Khzam, F.N.: An improved kernelization algorithm for r-set packing. Inf. Process. Lett. 110(16), 621–624 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abu-Khzam, F.N.: A kernelization algorithm for d-hitting set. J. Comput. Syst. Sci. 76(7), 524–531 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alber, J., Fernau, H., Niedermeier, R.: Parameterized complexity: exponential speed-up for planar graph problems. J. Algorithms 52(1), 26–56 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alber, J., Fiala, J.: Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J. Algorithms 52(2), 134–151 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alon, N., Lokshtanov, D., Saurabh, S.: Fast FAST. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 49–58. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  6. Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barát, J.: Directed path-width and monotonicity in digraph searching. Graphs and Combinatorics 22(2), 161–172 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Becker, A., Bar-Yehuda, R., Geiger, D.: Randomized algorithms for the loop cutset problem. J. Artif. Intell. Res. (JAIR) 12, 219–234 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Berwanger, D., Dawar, A., Hunter, P., Kreutzer, S.: DAG-Width and Parity Games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 524–536. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  10. Björklund, A.: Determinant sums for undirected hamiltonicity. In: Proceedings of the 51st Annual Symposium on Foundations of Computer Science, FOCS 2010, pp. 173–182 (2010)

    Google Scholar 

  11. Bodlaender, H.L.: On disjoint cycles. Int. J. Found. Comput. Sci. 5(1), 59–68 (1994)

    Article  MATH  Google Scholar 

  12. Bodlaender, H.L.: Kernelization: New Upper and Lower Bound Techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  13. Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) kernelization. In: Proceedings of the 50th Annual Symposium on Foundations of Computer Science, FOCS 2009, pp. 629–638 (2009)

    Google Scholar 

  15. Bodlaender, H.L., Heggernes, P., Villanger, Y.: Faster parameterized algorithms for minimum fill-in. Algorithmica 61(4), 817–838 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bousquet, N., Daligault, J., Thomassé, S.: Multicut is FPT. In: Proceedings of the 43rd Annual Symposium on Theory of Computing, STOC 2011, pp. 459–468 (2011)

    Google Scholar 

  17. Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inform. Process. Lett. 58(4), 171–176 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Calabro, C., Impagliazzo, R., Paturi, R.: The Complexity of Satisfiability of Small Depth Circuits. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 75–85. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  19. Cechlárová, K., Schlotter, I.: Computing the Deficiency of Housing Markets with Duplicate Houses. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 72–83. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  20. Chen, J., Fernau, H., Kanj, I.A., Xia, G.: Parametric duality and kernelization: Lower bounds and upper bounds on kernel size. SIAM J. Comput. 37(4), 1077–1106 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, J., Huang, X., Kanj, I.A., Xia, G.: Linear FPT reductions and computational lower bounds. In: Proceedings of the 36th Annual Symposium on Theory of Computing, STOC 2004, pp. 212–221. ACM, New York (2004)

    Google Scholar 

  23. Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411(40-42), 3736–3756 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chen, J., Liu, Y., Lu, S.: An Improved Parameterized Algorithm for the Minimum Node Multiway Cut Problem. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 495–506. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

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

    Google Scholar 

  26. Chitnis, R., Hajiaghayi, M., Marx, D.: Fixed-parameter tractability of directed multiway cut parameterized by the size of the cutset. In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pp. 1713–1725 (2012)

    Google Scholar 

  27. Chudnovsky, M., Fradkin, A.O., Seymour, P.D.: Tournament immersion and cutwidth. J. Comb. Theory, Ser. B 102(1), 93–101 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Chudnovsky, M., Scott, A., Seymour, P.: Vertex disjoint paths in tournaments (manuscript)

    Google Scholar 

  29. Cygan, M., Marx, D., Pilipczuk, M., Pilipczuk, M., Schlotter, I.: Parameterized Complexity of Eulerian Deletion Problems. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 131–142. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  30. Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: Proceedings of the 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, pp. 150–159 (2011)

    Google Scholar 

  31. Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: On Multiway Cut Parameterized above Lower Bounds. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 1–12. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  32. Dantsin, E., Hirsch, E.A.: Satisfiability Certificates Verifiable in Subexponential Time. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 19–32. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  33. Dehne, F., Fellows, M., Langston, M., Rosamond, F., Stevens, K.: An \(O(2\sp{O(k)}n\sp 3)\) FPT algorithm for the undirected feedback vertex set problem. Theory Comput. Syst. 41(3), 479–492 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Dell, H., Marx, D.: Kernelization of packing problems. In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pp. 68–81 (2012)

    Google Scholar 

  35. Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: Proceedings of the 42nd Annual Symposium on Theory of Computing, STOC 2010, pp. 251–260 (2010)

    Google Scholar 

  36. Demaine, E.D., Hajiaghayi, M.: The bidimensionality theory and its algorithmic applications. Comput. J. 51(3), 292–302 (2008)

    Article  Google Scholar 

  37. Demaine, E.D., Hajiaghayi, M., Thilikos, D.M.: The bidimensional theory of bounded-genus graphs. SIAM J. Discrete Math. 20(2), 357–371 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Demaine, E.D., Hajiaghayi, M.T.: Bidimensionality: new connections between FPT algorithms and PTASs. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 590–601 (2005)

    Google Scholar 

  39. Dorn, F.: Planar subgraph isomorphism revisited. In: Proceedings 27th International Symposium on Theoretical Aspects of Computer Science, STACS 2010. Dagstuhl Seminar Proceedings, vol. 5, pp. 263–274. Leibniz-Zentrum für Informatik, Schloss Dagstuhl, Germany (2010)

    Google Scholar 

  40. Dorn, F., Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S.: Beyond bidimensionality: Parameterized subexponential algorithms on directed graphs. In: Proceedings 27th International Symposium on Theoretical Aspects of Computer Science, STACS 2010, Dagstuhl Seminar Proceedings, vol. 5, pp. 251–262. Leibniz-Zentrum für Informatik, Schloss Dagstuhl, Germany (2010)

    Google Scholar 

  41. Downey, R.: A Basic Parameterized Complexity Primer. In: Bodlaender, H.L., et al. (eds.) Fellows Festschrift. LNCS, vol. 7370, pp. 91–128. Springer, Heidelberg (2012)

    Google Scholar 

  42. Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. In: Proceedings of the Twenty-first Manitoba Conference on Numerical Mathematics and Computing, Winnipeg, MB, vol. 87, pp. 161–178 (1992)

    Google Scholar 

  43. Downey, R.G., Fellows, M.R.: Parameterized computational feasibility. In: Clote, P., Remmel, J. (eds.) Proceedings of the Second Cornell Workshop on Feasible Mathematics, Feasible Mathematics II, pp. 219–244. Birkhäuser, Boston (1995)

    Chapter  Google Scholar 

  44. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)

    Google Scholar 

  45. Eppstein, D.: Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl. 3(3) (1999)

    Google Scholar 

  46. Feige, U.: Faster FAST (feedback arc set in tournaments). CoRR, abs/0911.5094 (2009)

    Google Scholar 

  47. Fellows, M.R., Fomin, F.V., Lokshtanov, D., Losievskaja, E., Rosamond, F.A., Saurabh, S.: Distortion Is Fixed Parameter Tractable. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 463–474. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  48. Fellows, M.R., Fomin, F.V., Lokshtanov, D., Rosamond, F.A., Saurabh, S., Szeider, S., Thomassen, C.: On the complexity of some colorful problems parameterized by treewidth. Inf. Comput. 209(2), 143–153 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  49. Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM J. Comput. 33(4), 892–922 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  50. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)

    MATH  Google Scholar 

  51. Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Algorithmic lower bounds for problems parameterized with clique-width. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 493–502 (2010)

    Google Scholar 

  52. Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms, 1st edn. Springer (2010)

    Google Scholar 

  53. Fomin, F.V., Pilipzuk, M.: Jungles, bundles, and fixed parameter tractability. CoRR, abs/1112. 1538 (2011)

    Google Scholar 

  54. Fomin, F.V., Villanger, Y.: Subexponential parameterized algorithm for minimum fill-in. In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pp. 1737–1746 (2012)

    Google Scholar 

  55. Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: Proceedings of the 40th Annual Symposium on Theory of Computing, STOC 2008, pp. 133–142 (2008)

    Google Scholar 

  56. Ganian, R., Hliněný, P., Kneis, J., Meister, D., Obdržálek, J., Rossmanith, P., Sikdar, S.: Are There Any Good Digraph Width Measures? In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 135–146. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  57. Gramm, J., Niedermeier, R., Rossmanith, P.: Fixed-parameter algorithms for closest string and related problems. Algorithmica 37(1), 25–42 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  58. Guillemot, S.: FPT algorithms for path-transversal and cycle-transversal problems. Discrete Optimization 8(1), 61–71 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  60. Hertli, T.: 3-SAT faster and simpler — Unique-SAT bounds for PPSZ hold in general. In: Proceedings of the 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, pp. 277–284 (2011)

    Google Scholar 

  61. Hüffner, F., Niedermeier, R., Wernicke, S.: Techniques for practical fixed-parameter algorithms. Comput. J. 51(1), 7–25 (2008)

    Article  Google Scholar 

  62. Hunter, P., Kreutzer, S.: Digraph measures: Kelly decompositions, games, and orderings. Theor. Comput. Sci. 399(3), 206–219 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  63. Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. System Sci. 63(4), 512–530 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  64. Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. J. Comb. Theory, Ser. B 82(1), 138–154 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  65. Kaplan, H., Shamir, R., Tarjan, R.E.: Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput. 28(5), 1906–1922 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  66. Kawarabayashi, K.: Planarity allowing few error vertices in linear time. In: Proceedings of the 50th Annual Symposium on Foundations of Computer Science, FOCS 2009, pp. 639–648 (2009)

    Google Scholar 

  67. Khot, S., Raman, V.: Parameterized complexity of finding subgraphs with hereditary properties. Theor. Comput. Sci. 289(2), 997–1008 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  68. Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2 − ε. In: 18th Annual IEEE Conference on Computational Complexity (CCC 2003), pp. 371–378 (2003)

    Google Scholar 

  69. Kratsch, S.: Co-nondeterminism in compositions: A kernelization lower bound for a Ramsey-type problem. In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pp. 114–122 (2012)

    Google Scholar 

  70. Kratsch, S., Pilipczuk, M., Pilipczuk, M., Wahlström, M.: Fixed-parameter tractability of multicut in directed acyclic graphs. CoRR, abs/1202.5749 (2012)

    Google Scholar 

  71. Kreutzer, S., Ordyniak, S.: Digraph decompositions and monotonicity in digraph searching. Theor. Comput. Sci. 412(35), 4688–4703 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  72. Kreutzer, S., Tazari, S.: Directed nowhere dense classes of graphs. In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pp. 1552–1562 (2012)

    Google Scholar 

  73. Langston, M.A.: Fixed-Parameter Tractability, A Prehistory. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) Fellows Festschrift. LNCS, vol. 7370, pp. 3–16. Springer, Heidelberg (2012)

    Google Scholar 

  74. Lokshtanov, D., Marx, D., Saurabh, S.: Known algorithms on graphs of bounded treewidth are probably optimal. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, pp. 777–789 (2011)

    Google Scholar 

  75. Lokshtanov, D., Marx, D., Saurabh, S.: Slightly superexponential parameterized problems. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, pp. 760–776 (2011)

    Google Scholar 

  76. Lokshtanov, D., Misra, N., Saurabh, S.: Kernelization – Preprocessing with a Guarantee. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) Fellows Festschrift. LNCS, vol. 7370, pp. 129–161. Springer, Heidelberg (2012)

    Google Scholar 

  77. Ma, B., Sun, X.: More efficient algorithms for closest string and substring problems. SIAM J. Comput. 39(4), 1432–1443 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  78. Marx, D.: Parameterized graph separation problems. Theoret. Comput. Sci. 351(3), 394–406 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  79. Marx, D.: On the optimality of planar and geometric approximation schemes. In: 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), pp. 338–348 (2007)

    Google Scholar 

  80. Marx, D.: Closest substring problems with small distances. SIAM Journal on Computing 38(4), 1382–1410 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  81. Marx, D.: Can you beat treewidth? Theory of Computing 6(1), 85–112 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  82. Marx, D., Razgon, I.: Fixed-parameter tractability of multicut parameterized by the size of the cutset. In: Proceedings of the 43rd Annual Symposium on Theory of Computing, STOC 2011, pp. 469–478 (2011)

    Google Scholar 

  83. Marx, D., Schlotter, I.: Obtaining a planar graph by vertex deletion. Algorithmica 62, 807–822 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  84. Misra, N., Raman, V., Saurabh, S.: Lower bounds on kernelization. Discrete Optimization 8(1), 110–128 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  85. Monien, B.: How to find long paths efficiently. In: Analysis and design of algorithms for combinatorial problems (Udine, 1982). North-Holland Math. Stud, vol. 109, pp. 239–254. North-Holland, Amsterdam (1985)

    Chapter  Google Scholar 

  86. Nemhauser, G.L., Trotter Jr., L.E.: Vertex packings: structural properties and algorithms. Math. Programming 8, 232–248 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  87. Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  88. Patrascu, M., Williams, R.: On the possibility of faster SAT algorithms. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 1065–1075 (2010)

    Google Scholar 

  89. Paturi, R., Pudlák, P.: On the complexity of circuit satisfiability. In: Proceedings of the 42nd Annual Symposium on Theory of Computing, STOC 2010, pp. 241–250 (2010)

    Google Scholar 

  90. Raman, V., Saurabh, S.: Parameterized algorithms for feedback set problems and their duals in tournaments. Theor. Comput. Sci. 351(3), 446–458 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  92. Reed, B., Smith, K., Vetta, A.: Finding odd cycle transversals. Operations Research Letters 32(4), 299–301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  93. Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Combin. Theory Ser. B 63(1), 65–110 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  94. Robertson, N., Seymour, P.D.: Graph minors. XX. Wagner’s conjecture. J. Combin. Theory Ser. B 92(2), 325–357 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  95. Scheffler, P.: A practical linear time algorithm for disjoint paths in graphs with bounded tree-width. Tech. Rep. 396/1994, Technical University of Berlin (1994)

    Google Scholar 

  96. Thilikos, D.M.: Graph Minors and Parameterized Algorithm Design. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) Fellows Festschrift. LNCS, vol. 7370, pp. 228–256. Springer, Heidelberg (2012)

    Google Scholar 

  97. Williams, R.: Finding paths of length k in O*(2k) time. Inf. Process. Lett. 109(6), 315–318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  98. Yap, C.K.: Some consequences of non-uniform conditions on uniform classes. Theor. Comput. Sci. 26, 287–300 (1983)

    Article  MATH  Google Scholar 

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  1. Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary

    Dániel Marx

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  1. Department of Information and Computing Sciences, Utrecht University, P.O. Box 80.089, 3508 TB, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. School of Mathematics, Statistics and Operations Research, Victoria University, P.O. Box 600, Wellington, New Zealand

    Rod Downey

  3. Institute of Informatics, University of Bergen, Postboks 7800, 5020, Bergen, Norway

    Fedor V. Fomin

  4. Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI), Pf. 63, 1518, Budapest, Hungary

    Dániel Marx

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Marx, D. (2012). What’s Next? Future Directions in Parameterized Complexity. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds) The Multivariate Algorithmic Revolution and Beyond. Lecture Notes in Computer Science, vol 7370. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30891-8_20

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