Iterative Compression for Exactly Solving NP-Hard Minimization Problems

  • Jiong Guo
  • Hannes Moser
  • Rolf Niedermeier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5515)

Abstract

We survey the conceptual framework and several applications of the iterative compression technique introduced in 2004 by Reed, Smith, and Vetta. This technique has proven very useful for achieving a number of recent breakthroughs in the development of fixed-parameter algorithms for NP-hard minimization problems. There is a clear potential for further applications as well as a further development of the technique itself. We describe several algorithmic results based on iterative compression and point out some challenges for future research.

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References

  1. 1.
    Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discrete Math. 3(2), 289–297 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Mach. Learn. 56(1-3), 89–113 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Becker, A., Bar-Yehuda, R., Geiger, D.: Randomized algorithms for the Loop Cutset problem. J. Artificial Intelligence Res. 12, 219–234 (2000)MathSciNetMATHGoogle Scholar
  4. 4.
    Becker, A., Geiger, D.: Approximation algorithms for the Loop Cutset problem. In: Proc. 10th UAI, pp. 60–68. Morgan Kaufmann, San Francisco (1994)Google Scholar
  5. 5.
    Betzler, N., Fellows, M.R., Guo, J., Niedermeier, R., Rosamond, F.A.: Fixed-parameter algorithms for Kemeny scores. In: Fleischer, R., Xu, J. (eds.) AAIM 2008. LNCS, vol. 5034, pp. 60–71. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Böckenhauer, H.-J., Hromkovic, J., Mömke, T., Widmayer, P.: On the hardness of reoptimization. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 50–65. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Bodlaender, H.L.: On disjoint cycles. Int. J. Found. Computer Science 5, 59–68 (1994)CrossRefMATHGoogle Scholar
  8. 8.
    Bodlaender, H.L.: A cubic kernel for feedback vertex set. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 320–331. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Chen, J., Fomin, F.V., Liu, Y., Lu, S., Villanger, Y.: Improved algorithms for the feedback vertex set problems. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 422–433. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. In: Proc. 40th STOC, pp. 177–186. ACM Press, New York (2008)Google Scholar
  12. 12.
    Dehne, F.K.H.A., Fellows, M.R., Langston, M.A., Rosamond, F.A., Stevens, K.: An O(2O(k) n 3) FPT algorithm for the undirected feedback vertex set problem. Theory Comput. Syst. 41(3), 479–492 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dehne, F.K.H.A., Fellows, M.R., Rosamond, F.A., Shaw, P.: 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: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 271–280. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Dom, M., Guo, J., Hüffner, F., Niedermeier, R., Truß, A.: Fixed-parameter tractability results for feedback set problems in tournaments. In: Calamoneri, T., Finocchi, I., Italiano, G.F. (eds.) CIAC 2006. LNCS, vol. 3998, pp. 320–331. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. Congr. Numer. 87, 161–187 (1992)MathSciNetMATHGoogle Scholar
  16. 16.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefMATHGoogle Scholar
  17. 17.
    Even, G., Naor, J., Schieber, B., Sudan, M.: Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 20(2), 151–174 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fellows, M.R., Hallett, M.T., Stege, U.: Analogs & duals of the MAST problem for sequences & trees. J. Algorithms 49(1), 192–216 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Festa, P., Pardalos, P.M., Resende, M.G.C.: Feedback set problems. In: Du, D.-Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, supplement vol. A, pp. 209–259. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar
  20. 20.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)MATHGoogle Scholar
  21. 21.
    Fomin, F., Gaspers, S., Kratsch, D., Liedloff, M., Saurabh, S.: Iterative compression and exact algorithms. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 335–346. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  22. 22.
    Fomin, F.V., Gaspers, S., Pyatkin, A.V.: Finding a minimum feedback vertex set in time O(1.7548n). In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 184–191. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  23. 23.
    Fulkerson, D.R., Ford Jr., L.R.: Maximal flow through a network. Canad. J. Math. 8, 399–404 (1956)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for network problems. SIAM J. Comput. 18(5), 1013–1036 (1989)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Multiway cuts in directed and node weighted graphs. In: Shamir, E., Abiteboul, S. (eds.) ICALP 1994. LNCS, vol. 820, pp. 487–498. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  26. 26.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-modeled data clustering: Exact algorithms for clique generation. Theory Comput. Syst. 38(4), 373–392 (2005)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Data reduction, exact, and heuristic algorithms for clique cover. In: Proc. 8th ALENEX, pp. 86–94. SIAM, Philadelphia (2006); Journal version to appear under the title Data reduction and exact algorithms for clique cover. ACM J. Exp. AlgorithmicsGoogle Scholar
  28. 28.
    Guillemot, S.: Parameterized complexity and approximability of the SLCS problem. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 115–128. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  29. 29.
    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)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Hüffner, F.: Algorithm engineering for optimal graph bipartization. In: Nikoletseas, S.E. (ed.) WEA 2005. LNCS, vol. 3503, pp. 240–252. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  31. 31.
    Hüffner, F.: Algorithms and Experiments for Parameterized Approaches to Hard Graph Problems. Ph.D thesis, Institut für Informatik, Friedrich-Schiller-Universität Jena (2007)Google Scholar
  32. 32.
    Hüffner, F., Betzler, N., Niedermeier, R.: Optimal edge deletions for signed graph balancing. In: Demetrescu, C. (ed.) WEA 2007. LNCS, vol. 4525, pp. 297–310. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  33. 33.
    Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Fixed-parameter algorithms for cluster vertex deletion. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 711–722. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  34. 34.
    Hüffner, F., Niedermeier, R., Wernicke, S.: Techniques for practical fixed-parameter algorithms. The Computer Journal 51(1), 7–25 (2008)CrossRefGoogle Scholar
  35. 35.
    Kahng, A.B., Vaya, S., Zelikovsky, A.: New graph bipartizations for double-exposure, bright field alternating phase-shift mask layout. In: Proc. Asia and South Pacific Design Automation Conference, pp. 133–138. ACM Press, New York (2001)Google Scholar
  36. 36.
    Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. J. Algorithms 31(2), 335–354 (1999)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Marx, D.: Chordal deletion is fixed-parameter tractable. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 37–48. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  38. 38.
    Mishra, S., Raman, V., Saurabh, S., Sikdar, S., Subramanian, C.R.: The complexity of finding subgraphs whose matching number equals the vertex cover number. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 268–279. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  39. 39.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefMATHGoogle Scholar
  40. 40.
    Pop, M., Kosack, D.S., Salzberg, S.L.: Hierarchical scaffolding with Bambus. Genome Research 14(1), 149–159 (2004)CrossRefGoogle Scholar
  41. 41.
    Raman, V., Saurabh, S., Sikdar, S.: Efficient exact algorithms through enumerating maximal independent sets and other techniques. Theory Comput. Syst. 41(3), 563–587 (2007)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Razgon, I., O’Sullivan, B.: Almost 2-SAT is fixed-parameter tractable. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 551–562. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  43. 43.
    Reed, B., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Schulz, A.S., Weismantel, R., Ziegler, G.M.: 0/1-integer programming: Optimization and augmentation are equivalent. In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 473–483, pp. 473–483. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  45. 45.
    Zhang, X.-S., Wang, R.-S., Wu, L.-Y., Chen, L.: Models and algorithms for haplotyping problem. Current Bioinformatics 1(1), 104–114 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jiong Guo
    • 1
  • Hannes Moser
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany

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