Ubiquitous Parameterization — Invitation to Fixed-Parameter Algorithms

  • Rolf Niedermeier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3153)

Abstract

Problem parameters are ubiquitous. In every area of computer science, we find all kinds of “special aspects” to the problems encountered. Hence, the study of parameterized complexity for computationally hard problems is proving highly fruitful. The purpose of this article is to stir the reader’s interest in this field by providing a gentle introduction to the rewarding field of fixed-parameter algorithms.

Keywords

NP-hardness parameterized complexity fixed-parameter algorithms parameterization 

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References

  1. 1.
    Abu-Khzam, F.N., Collins, R.L., Fellows, M.R., Langston, M.A., Suters, W.H., Symons, C.T.: Kernelization algorithms for the Vertex Cover problem: theory and experiments. In: Proc. ALENEX 2004, ACM/SIAM (2004)Google Scholar
  2. 2.
    Alber, J.: Exact Algorithms for NP-hard Problems on Networks: Design, Analysis, and Implementation. PhD thesis, WSI für Informatik, Universität Tübingen, Germany (2003) Google Scholar
  3. 3.
    Alber, J., Betzler, N., Niedermeier, R.: Experiments on data reduction for optimal domination in networks. In: Proc. International Network Optimization Conference (INOC 2003), Evry/Paris, France, October 2003, pp. 1–6 (2003)Google Scholar
  4. 4.
    Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for Dominating Set and related problems on planar graphs. Algorithmica 33(4), 461–493 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Alber, J., Dorn, F., Niedermeier, R.: Experimental evaluation of a tree decomposition based algorithm for Vertex Cover on planar graphs. Discrete Applied Mathematics (2004) (to appear)Google Scholar
  6. 6.
    Alber, J., Fan, H., Fellows, M.R., Fernau, H., Niedermeier, R., Rosamond, F., Stege, U.: Refined search tree technique for Dominating Set on planar graphs. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 111–122. Springer, Heidelberg (2001) Long version to appear in Journal of Computer and System SciencesGoogle Scholar
  7. 7.
    Alber, J., Fellows, M.R., Niedermeier, R.: Efficient data reduction for Dominating Set: A linear problem kernel for the planar case. In: Penttonen, M., Schmidt, E.M. (eds.) SWAT 2002. LNCS, vol. 2368, pp. 150–159. Springer, Heidelberg (2002) Long version to appear in Journal of the ACMGoogle Scholar
  8. 8.
    Alber, J., Fernau, H., Niedermeier, R.: Parameterized complexity: exponential speed-up for planar graph problems. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 261–272. Springer, Heidelberg (2001) Long version to appear in Journal of AlgorithmsGoogle Scholar
  9. 9.
    Alber, J., Fernau, H., Niedermeier, R.: Graph separators: A parameterized view. Journal of Computer and System Sciences 67(4), 808–832 (2003)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Alber, J., Gramm, J., Niedermeier, R.: Faster exact solutions for hard problems: a parameterized point of view. Discrete Mathematics 229, 3–27 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. Journal of the ACM 42(4), 844–856 (1995)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation — Combinatorial Optimization Problems and Their Approximability Properties. Springer, Heidelberg (1999)MATHGoogle Scholar
  13. 13.
    Bodlaender, H.L.: Treewidth: Algorithmic techniques and results. In: Privara, I., Ružička, P. (eds.) MFCS 1997. LNCS, vol. 1295, pp. 19–36. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  14. 14.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science 209, 1–45 (1998)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Wareham, H.T.: The parameterized complexity of sequence alignment and consensus. Theoretical Computer Science 147(1-2), 31–54 (1995)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Cai, L.: Parameterized complexity of Vertex Colouring. Discrete Applied Mathematics 127(1), 415–429 (2003)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Cai, L., Chen, J.: On fixed-parameter tractability and approximability of NP optimization problems. Journal of Computer and System Science 54(3), 465–474 (1997)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. Journal of Computer and System Sciences 67(4), 789–807 (2003)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Cesati, M., Trevisan, L.: On the efficiency of polynomial time approximation schemes. Information Processing Letters 64(4), 165–171 (1997)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Cheetham, J., Dehne, F., Rau-Chaplin, A., Stege, U., Taillon, P.J.: Solving large FPT problems on coarse-grained parallel machines. Journal of Computer and System Sciences 67(4), 691–706 (2003)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Chen, J., Huang, X., Kanj, I.A., Xia, G.: Strong computational lower bounds via parameterized complexity. In: Manuscript, a preliminary version appears in 36th ACM STOC 2004 (2004)Google Scholar
  22. 22.
    Chen, J., Kanj, I.A., Jia, W.: Vertex Cover: Further observations and further improvements. Journal of Algorithms 41, 280–301 (2001)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Dantsin, E., Hirsch, E.A., Ivanov, S., Vsemirnov, M.: Algorithms for SAT and upper bounds on their complexity. Technical Report TR01-012, Electronic Colloquium on Computational Complexity (2001)Google Scholar
  24. 24.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M.T., Thilikos, D.M.: Bidimensional parameters and local treewidth. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 109–118. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  25. 25.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M.T., Thilikos, D.M.: Subexponential parameterized algorithms on graphs of bounded-genus and H-minor-free graphs. In: Proc. 15th SODA, pp. 830–839. ACM/SIAM (2004)Google Scholar
  26. 26.
    Deng, X., Li, G., Li, Z., Ma, B., Wang, L.: Genetic design of drugs without side-effects. SIAM Journal on Computing 32(4), 1073–1090 (2003)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Downey, R.G.: Parameterized complexity for the skeptic. In: Proc. 18th IEEE Annual Conference on Computational Complexity, pp. 147–169 (2003)Google Scholar
  28. 28.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  29. 29.
    Downey, R.G., Fellows, M.R., Prieto-Rodriguez, E., Rosamond, F.A.: Fixedparameter tractability and completeness V: parametric miniatures. Manuscript (2003)Google Scholar
  30. 30.
    Dreyfus, S.E., Wagner, R.A.: The Steiner problem in graphs. Networks 1, 195–207 (1972)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Ellis, J., Fan, H., Fellows, M.R.: The Dominating Set problem is fixed parameter tractable for graphs of bounded genus. In: Penttonen, M., Schmidt, E.M. (eds.) SWAT 2002. LNCS, vol. 2368, pp. 180–189. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  32. 32.
    Fedin, S.S., Kulikov, A.S.: Automated proofs of upper bounds on the running time of splitting algorithms. Manuscript (September 2003)Google Scholar
  33. 33.
    Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45(4), 634–652 (1998)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Fellows, M.R.: Parameterized complexity: The main ideas and connections to practical computing. In: Fleischer, R., Moret, B.M.E., Schmidt, E.M. (eds.) Experimental Algorithmics. LNCS, vol. 2547, pp. 51–77. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  35. 35.
    Fellows, M.R.: Blow-ups, win/win’s, and crown rules: Some new directions in FPT. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 1–12. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  36. 36.
    Fellows, M.R.: New directions and new challenges in algorithm design and complexity, parameterized. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 505–520. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  37. 37.
    Fellows, M.R., Gramm, J., Niedermeier, R.: On the parameterized intractability of Closest Substring and related problems. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 262–273. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  38. 38.
    Fernau, H.: Parametric duality: kernel sizes & algorithmics. Technical Report TR04-027, Electronic Colloquium on Computational Complexity (2004) Google Scholar
  39. 39.
    Flum, J., Grohe, M.: The parameterized complexity of counting problems. In: Proc. 43rd FOCS, pp. 538–550. IEEE Computer Society, Los Alamitos (2002)Google Scholar
  40. 40.
    Flum, J., Grohe, M., Weyer, M.: Bounded fixed-parameter tractability and log2n nondeterministic bits. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 555–567. Springer, Heidelberg (2004) (to appear)CrossRefGoogle Scholar
  41. 41.
    Fomin, F.V., Thilikos, D.M.: Dominating sets in planar graphs: branch-width and exponential speed-up. In: Proc. 14th SODA, pp. 168–177. ACM/SIAM (2003)Google Scholar
  42. 42.
    Fomin, F.V., Thilikos, D.M.: A simple and fast approach for solving problems on planar graphs. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 56–67. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  43. 43.
    Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. In: Proc. Logic in Computer Science, pp. 215–224. IEEE Computer Society, Los Alamitos (2002)Google Scholar
  44. 44.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)MATHGoogle Scholar
  45. 45.
    Garg, N., Vazirani, V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18(1), 3–20 (1997)MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Gramm, J.: Fixed-Parameter Algorithms for the Consensus Analysis of Genomic Sequences. PhD thesis, WSI für Informatik, Universität Tübingen, Germany (2003) Google Scholar
  47. 47.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Automated generation of search tree algorithms for hard graph modification problems. Algorithmica 39(4), 321–347 (2004)MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Gramm, J., Guo, J., Niedermeier, R.: On exact and approximation algorithms for Distinguishing Substring Selection. In: Lingas, A., Nilsson, B.J. (eds.) FCT 2003. LNCS, vol. 2751, pp. 261–272. Springer, Heidelberg (2003) Long version to appear in Theory of Computing SystemsGoogle Scholar
  49. 49.
    Gramm, J., Niedermeier, R.: Breakpoint medians and breakpoint phylogenies: a fixed-parameter approach. Bioinformatics 18(Suppl. 2), S128–S139 (2002)Google Scholar
  50. 50.
    Gramm, J., Niedermeier, R.: A fixed-parameter algorithm for Minimum Quartet Inconsistency. Journal of Computer and System Sciences 67(4), 723–741 (2003)MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Gramm, J., Niedermeier, R., Rossmanith, P.: Fixed-parameter algorithms for Closest String and related problems. Algorithmica 37(1), 25–42 (2003)MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Grohe, M.: Parameterized complexity for the database theorist. SIGMOD Record 31(4), 86–96 (2002)CrossRefGoogle Scholar
  53. 53.
    Guo, J., Hüffner, F., Niedermeier, R.: A structural view on parameterizing problems: distance from triviality. Manuscript (April 2004)Google Scholar
  54. 54.
    Guo, J., Niedermeier, R.: Exact algorithms for Tree-like Weighted Set Cover. Manuscript (April 2004)Google Scholar
  55. 55.
    Guo, J., Niedermeier, R.: Fixed-parameter tractability results for Mulitcut in Trees. Manuscript (May 2004)Google Scholar
  56. 56.
    Hirsch, E.A.: New worst-case upper bounds for SAT. Journal of Automated Reasoning 24(4), 397–420 (2000)MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Hochbaum, D. (ed.): Approximation Algorithms for NP-hard Problems. PWS Publishing Company (1997)Google Scholar
  58. 58.
    Hoffmann, M., Okamoto, Y.: The traveling salesman problem with few inner points. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 268–277. Springer, Heidelberg (2004) (to appear)Google Scholar
  59. 59.
    Hofri, M.: Analysis of Algorithms: Computational Methods and Mathematical Tools. Oxford University Press, Oxford (1995)MATHGoogle Scholar
  60. 60.
    Hromkovič, J.: Algorithmics for Hard Problems. Springer, Heidelberg (2002)Google Scholar
  61. 61.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? Journal of Computer and System Sciences 63(4), 512–530 (2001)MATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Iwama, K., Tamaki, S.: Improved upper bounds for 3-SAT. Technical Report TR03-053, Electronic Colloquium on Computational Complexity (2003), Also appears in Proc. ACM/SIAM SODA 2004Google Scholar
  63. 63.
    Juedes, D., Chor, B., Fellows, M.R.: Linear kernels in linear time, or how to save k colors in o(n2) steps. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 257–269. Springer, Heidelberg (2004) (to appear)Google Scholar
  64. 64.
    Khuller, S.: The Vertex Cover problem. SIGACT News 33(2), 31–33 (2002)CrossRefMathSciNetGoogle Scholar
  65. 65.
    Küchlin, W., Sinz, C.: Proving consistency assertions for automotive product data management. Journal of Automated Reasoning 24(1-2), 145–163 (2000)MATHCrossRefGoogle Scholar
  66. 66.
    Kullmann, O.: New methods for 3-SAT decision and worst-case analysis. Theoretical Computer Science 223(1-2), 1–72 (1999)MATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    Li, M., Ma, B., Wang, L.: On the Closest String and Substring problems. Journal of the ACM 49(2), 157–171 (2002)CrossRefMathSciNetGoogle Scholar
  68. 68.
    Lipton, R., Tarjan, R.: Applications of a planar separator theorem. SIAM Journal on Computing 9(3), 615–627 (1980)MATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. Journal of Algorithms 31(2), 335–354 (1999)MATHCrossRefMathSciNetGoogle Scholar
  70. 70.
    McCartin, C.: Parameterized counting problems. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 556–567. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  71. 71.
    Michalewicz, Z., Fogel, B.F.: How to Solve it: Modern Heuristics. Springer, Heidelberg (2000)MATHGoogle Scholar
  72. 72.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  73. 73.
    Nemhauser, G.L., Trotter, J.L.E.: Vertex packing: structural properties and algorithms. Mathematical Programming 8, 232–248 (1975)MATHCrossRefMathSciNetGoogle Scholar
  74. 74.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2005) (forthcoming)Google Scholar
  75. 75.
    Niedermeier, R., Rossmanith, P.: A general method to speed up fixed-parametertractable algorithms. Information Processing Letters 73, 125–129 (2000)MATHCrossRefMathSciNetGoogle Scholar
  76. 76.
    Niedermeier, R., Rossmanith, P.: On efficient fixed-parameter algorithms for Weighted Vertex Cover. Journal of Algorithms 47(2), 63–77 (2003)MATHMathSciNetGoogle Scholar
  77. 77.
    Nikolenko, S.I., Sirotkin, A.V.: Worst-case upper bounds for SAT: automated proof. In: 15th European Summer School in Logic Language and Information, ESSLLI 2003 (2003)Google Scholar
  78. 78.
    Pearl, J.: Heuristics. Addison–Wesley, Reading (1984)Google Scholar
  79. 79.
    Pietrzak, K.: On the parameterized complexity of the fixed alphabet Shortest Common Supersequence and Longest Common Subsequence problems. Journal of Computer and System Sciences 67(4), 757–771 (2003)MATHCrossRefMathSciNetGoogle Scholar
  80. 80.
    Sinz, C.: Visualizing the internal structure of SAT instances (preliminary report). In: Proc. 7th Intl. Conf. on Theory and Applications of Satisfiability Testing (SAT 2004), Vancouver, Canada (May 2004)Google Scholar
  81. 81.
    Sinz, C., Kaiser, A., Küchlin, W.: Formal methods for the validation of automotive product configuration data. Artificial Intelligence for Engineering Design, Analysis and Manufacturing 17(1), 75–97 (2003)CrossRefGoogle Scholar
  82. 82.
    Stearns, R.E., Hunt III, H.B.: Power indices and easier hard problems. Mathematical Systems Theory 23, 209–225 (1990)MATHCrossRefMathSciNetGoogle Scholar
  83. 83.
    Szeider, S.: Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 548–558. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  84. 84.
    Szeider, S.: On fixed-parameter tractable parameterizations of SAT. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 188–202. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  85. 85.
    Telle, J.A., Proskurowski, A.: Practical algorithms on partial k-trees with an application to domination-like problems. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1993. LNCS, vol. 709, pp. 610–621. Springer, Heidelberg (1993)Google Scholar
  86. 86.
    Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2001)Google Scholar
  87. 87.
    Woeginger, G.J.: Exact algorithms for NP-hard problems: A survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Rolf Niedermeier
    • 1
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany

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