Further Reflections on a Theory for Basic Algorithms

  • Allan Borodin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4041)


Can we optimally solve Max2SAT in (say) time (|F| log|F|) where |F| is the length of formula F. Of course, since Max2SAT is NP-complete, we can confidently rely on our strongly held belief that no NP-hard problem can be solved optimally in polynomial time. But obtaining unconditional complexity lower bounds (even linear or near linear bounds) remains the central challenge of complexity theory. In the complementary fields of complexity theory and that of algorithm design and analysis, we ask questions such as “what is the best polynomial time approximation ratio” that can be achieved for Max2SAT. The best negative results are derived from the beautiful development of PCP proofs. In terms of obtaining better approximation algorithms, we appeal to a variety of algorithmic techniques, including very basic techniques such as greedy algorithms, dynamic programming (with scaling), divide and conquer, local search and some more technically involved methods such as LP relaxation and randomized rounding, semi-definite programming (see [34] and [30] for an elegant presentation of these randomized methods and the concept of derandomization using conditional expectations). A more refined question might ask “what is the best approximation ratio (for a given problem such as Max2SAT) that can be obtained in (say) time O(n logn)” where n is the length of the input in some standard representation of the problem. What algorithmic techniques should we consider if we are constrained to time O(n logn)?


Dynamic Programming Approximation Algorithm Greedy Algorithm Approximation Ratio Online 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.
    Agrawal, A., Klein, P., Ravi, R.: When trees collide: An approximation algorithm for the generalized steiner problem on networks. SICOMP 24, 440–465 (1995)MATHMathSciNetGoogle Scholar
  2. 2.
    Angelopoulos, S.: Randomized priority algorithms. In: Solis-Oba, R., Jansen, K. (eds.) WAOA 2003. LNCS, vol. 2909, pp. 27–40. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Angelopoulos, S., Borodin, A.: On the power of priority algorithms for facility location and set cover. Algorithmica 40(4), 271–291 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Arora, S., Bollobás, B., Lovász, L.: Proving integrality gaps without knowing the linear program. In: Proceedings of the 43rd Annual IEEE Conference on Foundations of Computer Science, pp. 313–322 (2002)Google Scholar
  5. 5.
    Bar-Yehuda, R., Bendel, A., Freund, A., Rawitz, D.: Local ratio: A unified framework for approxmation algorithms in memoriam: Shimon even 1935-2004. Computing Surveys 36, 422–463 (2004)CrossRefGoogle Scholar
  6. 6.
    Bar-Yehuda, R., Rawitz, D.: On the equivalence between the primal-dual schema and the local ratio technique. In: 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX, pp. 24–35 (2001)Google Scholar
  7. 7.
    Borodin, A., Cashman, D., Magen, A.: How well can primal-dual and local-ratio algorithms perform? In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 943–955. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Borodin, A., Nielsen, M.N., Rackoff, C.: Incremental priority algorithms. Algorithmica 37(4), 295–326 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Borodin, A., Boyar, J., Larsen, K.S.: Priority Algorithms for Graph Optimization Problems. In: Persiano, G., Solis-Oba, R. (eds.) WAOA 2004. LNCS, vol. 3351, pp. 126–139. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Alekhnovich, M., Borodin, A., Buresh-Oppenheim, J., Impagliazzo, R., Magen, A., Pitassi, T.: Toward a model for backtracking and dynamic programming. In: Proceedings of Computational Complexity Conference (CCC), pp. 308–322 (2005)Google Scholar
  11. 11.
    Buresh-Oppenheim, J., Davis, S., Impagliazzo, R.: A formal model of dynamic programming algorithms (manuscript in preparation, 2006)Google Scholar
  12. 12.
    Chvátal, V.: A greedy heuristic for the set covering problem. Mathematics of Operations Research 4(3), 233–235 (1979)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Clarkson, K.L.: A modification of the greedy algorithm for vertex cover. Information Processing Letters 16, 23–25 (1983)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)MATHGoogle Scholar
  15. 15.
    Davis, S., Impagliazzo, R.: Models of greedy algorithms for graph problems. In: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (2004)Google Scholar
  16. 16.
    Edmonds, J.: Matroids and the greedy algorithm. Mathematical Programming 1, 127–136 (1971)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Erlebach, T., Spieksma, F.C.R.: Interval selection: Applications, algorithms, and lower bounds. Technical Report 152, Computer Engineering and Networks Laboratory, ETH (October 2002)Google Scholar
  18. 18.
    Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SICOMP 24, 296–317 (1995)MATHMathSciNetGoogle Scholar
  19. 19.
    Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms, pp. 649–657 (1998)Google Scholar
  20. 20.
    Helman, P.: A common schema for dynamic programming and branch and bound algorithms. Journal of the Association of Computing Machinery 36(1), 97–128 (1989)MATHMathSciNetGoogle Scholar
  21. 21.
    Horn, S.L.: One-pass algorithms with revocable acceptances for job interval selection. MSc Thesis, University of Toronto (2004)Google Scholar
  22. 22.
    Ibarra, O., Kim, C.: Fast approximation algorithms for the knapsack and sum of subset problems. JACM 4, 463–468 (1975)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and lagrangian relaxation. Journal of the ACM 48(2), 274–296 (2001)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Khanna, S., Motwani, R., Sudan, M., Vazirani, U.: On syntactic versus computational views of approximability. SIAM Journal on Computing 28, 164–191 (1998)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Korte, B., Lovász, L.: Mathematical structures underlying greedy algorithms. In: Gecseg, F. (ed.) FCT 1981. LNCS, vol. 117, pp. 205–209. Springer, Heidelberg (1981)Google Scholar
  26. 26.
    Lawler, E.L.: Fast approximation algorithms for knapsack problems. In: Proc. 18th Ann. Symp. on Foundations of Computer Science, Long Beach, CA. IEEE Computer Society Press, Los Alamitos (1977)Google Scholar
  27. 27.
    Mahdian, M., Ye, J., Zhang, J.: Improved approximation algorithms for metric facility location problems. In: Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pp. 229–242 (2002)Google Scholar
  28. 28.
    Marchetti-Spaccamela, A.: Personal communication as stated in [10] (2004)Google Scholar
  29. 29.
    Mestre, J.: Greedy in approximation algorithms (unpublished manuscript, 2006)Google Scholar
  30. 30.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  31. 31.
    Rado, R.: A theorem on independence relations. Quart. Jorunal of Mathematics 13, 83–89 (1942)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Regev, O.: Priority algorithms for makespan minimization in the subset model. Information Processing Letters 84(3), 153–157 (2002)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Sahni, S.: Approximate algorithms for the 0-1 knapsack problem. JACM 1, 115–124 (1975)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Vazirani, V.V.: Approximation algorithms. Springer, New York (2001)Google Scholar
  35. 35.
    Woeginger, G.: When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)? INFORMS Journal on Computing 12, 57–75 (2000)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Allan Borodin
    • 1
  1. 1.Department of Computer ScienceUniversity of Toronto 

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