We are reviewing recent advances in the run time analysis of search tree algorithms, including indications to open problems. In doing so, we also try to cover the historical dimensions of this topic.


Search Tree Travel Salesman Problem Travel Salesman Problem Edge Cover Search Tree Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study. Princeton Univ. Press, Princeton (2006)zbMATHGoogle Scholar
  2. 2.
    Bailey, M., Alden, J., Smith, R.L.: Approximate dynamic programming using epsilon-pruning (working paper). TR, University of Michigan, Ann Arbor (2002)Google Scholar
  3. 3.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: The Travelling Salesman Problem in bounded degree graphs. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 198–209. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Chen, J., Kanj, I.A., Xia, G.: Labeled search trees and amortized analysis: improved upper bounds for NP-hard problems. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 148–157. Springer, Heidelberg (2003)Google Scholar
  5. 5.
    Dube, S.: Using fractal geometry for solving divide-and-conquer recurrences (extended abstract). In: Ng, K.W., Balasubramanian, N.V., Raghavan, P., Chin, F.Y.L. (eds.) ISAAC 1993. LNCS, vol. 762, pp. 191–200. Springer, Heidelberg (1993)Google Scholar
  6. 6.
    Eppstein, D.: Quasiconvex analysis of multivariate recurrence equations for back tracking algorithms. ACM Trans. Algorithms 2, 492–509 (2006)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Fedin, S.S., Kulikov, A.S.: Automated proofs of upper bounds on the running time of splitting algorithms. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 248–259. Springer, Heidelberg (2004)Google Scholar
  8. 8.
    Feigenbaum, J., Kannan, S., Vardi, M.Y., Viswanathan, M.: The complexity of problems on graphs represented as OBDDs. Chicago J. Theor. Comput. Sci. (1999)Google Scholar
  9. 9.
    Fernau, H.: Two-layer planarization: improving on parameterized algorithmics. J. Graph Algorithms and Applications 9, 205–238 (2005)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Fernau, H.: Parameterized algorithms for hitting set: the weighted case. In: Calamoneri, T., Finocchi, I., Italiano, G.F. (eds.) CIAC 2006. LNCS, vol. 3998, pp. 332–343. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Fernau, H.: A top-down approach to search trees: improved algorithmics for 3-hitting set. Algorithmica (to appear)Google Scholar
  12. 12.
    Fernau, H., Raible, D.: Exact algorithms for maximum acyclic subgraph on a superclass of cubic graphs. In: Nakano, S.-i., Rahman, M. S. (eds.) WALCOM 2008. LNCS, vol. 4921, pp. 144–156. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge Univ. Press, Cambridge (2008)Google Scholar
  14. 14.
    Fomin, F.V., Gaspers, S., Pyatkin, A.V., Razgon, I.: On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms. Algorithmica 52(2), 293–307 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Fomin, F., Golovach, P., Kratsch, D., Kratochvil, J., Liedloff, M.: Branch & recharge: Exact algorithms for generalized domination. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 507–518. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: domination – a case study. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 191–203. Springer, Heidelberg (2005)Google Scholar
  17. 17.
    Fomin, F.V., Grandoni, F., Kratsch, D.: Solving connected dominating set faster than 2n. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 152–163. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: a simple O(20.288n) independent set algorithm. In: SODA, pp. 18–25 (2006)Google Scholar
  19. 19.
    Fomin, F.V., Grandoni, F., Pyatkin, A.V., Stepanov, A.A.: Combinatorial bounds via measure and conquer: Bounding minimal dominating sets and applications. ACM Trans. Algorithms 5, 1–17 (2008)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Gaspers, S.: Exponential Time Algorithms: Structures, Measures, and Bounds. Ph.D thesis, University of Bergen, Norway (2008)Google Scholar
  21. 21.
    Gaspers, S., Liedloff, M.: A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set in Graphs. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 78–89. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  22. 22.
    Golomb, S.W., Baumert, L.D.: Backtrack programming. J. ACM 12, 516–524 (1965)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Graham, R., Knuth, D.E., Patashnik, O.: Concrete Mathematics. Addison-Wesley, Reading (1989)zbMATHGoogle Scholar
  24. 24.
    Gramm, J.: Fixed-Parameter Algorithms for the Consensus Analysis of Genomic Data. Dissertation, Univ. Tübingen, Germany (2003)Google Scholar
  25. 25.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Automated generation of search tree algorithms for hard graph modification problems. Algorithmica 39, 321–347 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Hong, S.: A Linear Programming Approach for the Traveling Salesman Problem. Ph.D thesis, The Johns Hopkins University, Baltimore, Maryland, USA (1972)Google Scholar
  27. 27.
    Ibaraki, T.: Theoretical comparison of search strategies in branch-and-bound algorithms. Intern. J. Computer and Information Sciences 5, 315–344 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Ibaraki, T.: On the computational efficiency of branch-and-bound algorithms. J. Operations Research Society of Japan 26, 16–35 (1977)MathSciNetGoogle Scholar
  29. 29.
    Ibaraki, T.: The power of dominance relations in branch-and-bound algorithms. J. ACM 24, 264–279 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Ibaraki, T.: Branch-and-bound procedure and state-space representation of combinatorial optimization problems. Information and Control 36, 1–27 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Jeroslaw, R.G.: Trivial integer programs unsolvable by branch-and-bound. Mathematical Programming 6, 105–109 (1974)CrossRefGoogle Scholar
  32. 32.
    Jünger, M., Thienel, S.: The ABACUS system for branch-and-cut-and-price algorithms in integer programming and combinatorial optimization. Software: Practice and Experience 30, 1325–1352 (2000)zbMATHCrossRefGoogle Scholar
  33. 33.
    Karp, R.M., Held, M.: Finite-state processes and dynamic programming. SIAM J. Applied Mathematics 15, 693–718 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Knuth, D.E.: Estimating the efficiency of backtrack programs. Mathematics of Computation 29, 121–136 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Kulikov, A.S.: Automated generation of simplification rules for SAT and MAXSAT. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 430–436. Springer, Heidelberg (2005)Google Scholar
  36. 36.
    Kullmann, O.: New methods for 3-SAT decision and worst-case analysis. Theoretical Computer Science 223, 1–72 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Kullmann, O.: Fundaments of Branching Heuristics. In: Handbook of Satisfiability, ch. 7, pp. 205–244. IOS Press, Amsterdam (2009)Google Scholar
  38. 38.
    McDiarmid, C.J.H., Provan, G.M.A.: An expected-cost analysis of backtracking and non-backtracking algorithms. In: IJCAI 1991, vol. 1, pp. 172–177 (1991)Google Scholar
  39. 39.
    Mehlhorn, K.: Graph Algorithms and NP-Completeness. Springer, Heidelberg (1984)zbMATHGoogle Scholar
  40. 40.
    Mitten, L.G.: Branch-and-bound methods: general formulation and properties. Operations Research 18, 24–34 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Morin, T.L., Marsten, R.E.: Branch-and-bound strategies for dynamic programming. Operations Research 24, 611–627 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Univ. Press, Oxford (2006)zbMATHCrossRefGoogle Scholar
  43. 43.
    Raible, D., Fernau, H.: A new upper bound for max-2-sat: A graph-theoretic approach. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 551–562. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  44. 44.
    Raible, D., Fernau, H.: Power domination in O *(1.7548n) using reference search trees. In: ISAAC. LNCS, vol. 5369, pp. 136–147. Springer, Heidelberg (2008)Google Scholar
  45. 45.
    Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32, 57–95 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    van Rooij, J.M.M., Bodlaender, H.L.: Design by measure and conquer, a faster exact algorithm for dominating set. In: STACS, pp. 657–668. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl (2008)Google Scholar
  47. 47.
    Wahlström, M.: Algorithms, Measures and Upper Bounds for Satisfiability and Related Problems. Ph.D thesis, Linköpings universitet, Sweden (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Henning Fernau
    • 1
  • Daniel Raible
    • 1
  1. 1.Univ.Trier, FB 4—Abteilung InformatikTrierGermany

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