Approximation Algorithms

Chapter
Part of the Algorithms and Combinatorics book series (AC, volume 21)

Abstract

In this chapter we introduce the important concept of approximation algorithms. So far we have dealt mostly with polynomially solvable problems. In the remaining chapters we shall indicate some strategies to cope with NP-hard combinatorial optimization problems. Here approximation algorithms must be mentioned in the first place.

Keywords

Approximation Algorithm Chromatic Number Vertex Cover Truth Assignment Perfect Graph 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Asano, T., Iwama, K., Takada, H., and Yamashita, Y. [2000]: Designing high-quality approximation algorithms for combinatorial optimization problems. IEICE Transactions on Communications/Electronics/Information and Systems E83-D (2000), 462–478Google Scholar
  2. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., and Protasi, M. [1999]: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, Berlin 1999MATHGoogle Scholar
  3. Garey, M.R., and Johnson, D.S. [1979]: Computers and Intractability; A Guide to the Theory of NP-Completeness. Freeman, San Francisco 1979, Chapter 4Google Scholar
  4. Hochbaum, D.S. [1996]: Approximation Algorithms for NP-Hard Problems. PWS, Boston, 1996Google Scholar
  5. Horowitz, E., and Sahni, S. [1978]: Fundamentals of Computer Algorithms. Computer Science Press, Potomac 1978, Chapter 12Google Scholar
  6. Shmoys, D.B. [1995]: Computing near-optimal solutions to combinatorial optimization problems. In: Combinatorial Optimization; DIMACS Series in Discrete Mathematics and Theoretical Computer Science 20 (W. Cook, L. Lovász, P. Seymour, eds.), AMS, Providence 1995Google Scholar
  7. Papadimitriou, C.H. [1994]: Computational Complexity, Addison-Wesley, Reading 1994, Chapter 13Google Scholar
  8. Vazirani, V.V. [2001]: Approximation Algorithms. Springer, Berlin, 2001Google Scholar
  9. Williamson, D.P., and Shmoys, D.B. [2011]: The Design of Approximation Algorithms. Cambridge University Press, Cambridge 2011MATHGoogle Scholar
  10. Ajtai, M. [1994]: Recursive construction for 3-regular expanders. Combinatorica 14 (1994), 379–416CrossRefMATHMathSciNetGoogle Scholar
  11. Alizadeh, F. [1995]: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal on Optimization 5 (1995), 13–51CrossRefMATHMathSciNetGoogle Scholar
  12. Appel, K., and Haken, W. [1977]: Every planar map is four colorable; Part I; Discharging. Illinois Journal of Mathematics 21 (1977), 429–490MATHMathSciNetGoogle Scholar
  13. Appel, K., Haken, W., and Koch, J. [1977]: Every planar map is four colorable; Part II; Reducibility. Illinois Journal of Mathematics 21 (1977), 491–567MATHMathSciNetGoogle Scholar
  14. Arora, S. [1994]: Probabilistic checking of proofs and the hardness of approximation problems, Ph.D. thesis, U.C. Berkeley, 1994Google Scholar
  15. Arora, S., Lund, C., Motwani, R., Sudan, M., and Szegedy, M. [1998]: Proof verification and hardness of approximation problems. Journal of the ACM 45 (1998), 501–555CrossRefMATHMathSciNetGoogle Scholar
  16. Arora, S., and Safra, S. [1998]: Probabilistic checking of proofs. Journal of the ACM 45 (1998), 70–122CrossRefMATHMathSciNetGoogle Scholar
  17. Asano, T. [2006]: An improved analysis of Goemans and Williamson’s LP-relaxation for MAX SAT. Theoretical Computer Science 354 (2006), 339–353CrossRefMATHMathSciNetGoogle Scholar
  18. Avidor, A., Berkovitch, I., and Zwick, U. [2006]: Improved approximation algorithms for MAX NAE-SAT and MAX SAT. Approximation and Online Algorithms – Proceedings of the 3rd WAOA workshop (2005); LNCS 3879 (Erlebach, T., Persiano, G., eds.), Springer, Berlin 2006, pp. 27–40Google Scholar
  19. Bar-Yehuda, R., and Even, S. [1981]: A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms 2 (1981), 198–203CrossRefMATHMathSciNetGoogle Scholar
  20. Becker, A., and Geiger, D. [1996]: Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artificial Intelligence Journal 83 (1996), 1–22CrossRefMathSciNetGoogle Scholar
  21. Bellare, M., and Sudan, M. [1994]: Improved non-approximability results. Proceedings of the 26th Annual ACM Symposium on the Theory of Computing (1994), 184–193Google Scholar
  22. Bellare, M., Goldreich, O., and Sudan, M. [1998]: Free bits, PCPs and nonapproximability – towards tight results. SIAM Journal on Computing 27 (1998), 804–915CrossRefMATHMathSciNetGoogle Scholar
  23. Berge, C. [1961]: Färbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind. Wissenschaftliche Zeitschrift, Martin Luther Universität Halle-Wittenberg, Mathematisch-Naturwissenschaftliche Reihe (1961), 114–115Google Scholar
  24. Berge, C. [1962]: Sur une conjecture relative au problème des codes optimaux. Communication, 13ème assemblée générale de l’URSI, Tokyo 1962Google Scholar
  25. Berman, P., and Fujito, T. [1999]: On approximation properties of the independent set problem for low degree graphs. Theory of Computing Systems 32 (1999), 115–132CrossRefMATHMathSciNetGoogle Scholar
  26. Brooks, R.L. [1941]: On colouring the nodes of a network. Proceedings of the Cambridge Philosophical Society 37 (1941), 194–197CrossRefMathSciNetGoogle Scholar
  27. Chen, J., Friesen, D.K., and Zheng, H. [1999]: Tight bound on Johnson’s algorithm for maximum satisfiability. Journal of Computer and System Sciences 58 (1999), 622–640CrossRefMATHMathSciNetGoogle Scholar
  28. Chlebík, M. and Chlebíková, J. [2006]: Complexity of approximating bounded variants of optimization problems. Theoretical Computer Science 354 (2006), 320-338CrossRefMATHMathSciNetGoogle Scholar
  29. Chudnovsky, M., Cornuéjols, G., Liu, X., Seymour, P., and Vus̆ković, K. [2005]: Recognizing Berge graphs. Combinatorica 25 (2005), 143–186Google Scholar
  30. Chudnovsky, M., Robertson, N., Seymour, P., and Thomas, R. [2006]: The strong perfect graph theorem. Annals of Mathematics 164 (2006), 51–229CrossRefMATHMathSciNetGoogle Scholar
  31. Chvátal, V. [1975]: On certain polytopes associated with graphs. Journal of Combinatorial Theory B 18 (1975), 138–154CrossRefMATHGoogle Scholar
  32. Chvátal, V. [1979]: A greedy heuristic for the set cover problem. Mathematics of Operations Research 4 (1979), 233–235CrossRefMATHMathSciNetGoogle Scholar
  33. Clementi, A.E.F., and Trevisan, L. [1999]: Improved non-approximability results for minimum vertex cover with density constraints. Theoretical Computer Science 225 (1999), 113–128CrossRefMATHMathSciNetGoogle Scholar
  34. Deza, M.M., and Laurent, M. [1997]: Geometry of Cuts and Metrics. Springer, Berlin 1997MATHGoogle Scholar
  35. Dinur, I. [2007]: The PCP theorem by gap amplification. Journal of the ACM 54 (2007), Article 12Google Scholar
  36. Dinur, I., and Safra, S. [2002]: On the hardness of approximating minimum vertex cover. Annals of Mathematics 162 (2005), 439–485CrossRefMATHMathSciNetGoogle Scholar
  37. Erdős, P. [1967]: On bipartite subgraphs of graphs. Mat. Lapok. 18 (1967), 283–288MathSciNetGoogle Scholar
  38. Feige, U. [1998]: A threshold of lnn for approximating set cover. Journal of the ACM 45 (1998), 634–652CrossRefMATHMathSciNetGoogle Scholar
  39. Feige, U. [2004]: Approximating maximum clique by removing subgraphs. SIAM Journal on Discrete Mathematics 18 (2004), 219–225CrossRefMATHMathSciNetGoogle Scholar
  40. Feige, U., and Goemans, M.X. [1995]: Approximating the value of two prover proof systems, with applications to MAX 2SAT and MAX DICUT. Proceedings of the 3rd Israel Symposium on Theory of Computing and Systems (1995), 182–189Google Scholar
  41. Feige, U., Goldwasser, S., Lovász, L., Safra, S., and Szegedy, M. [1996]: Interactive proofs and the hardness of approximating cliques. Journal of the ACM 43 (1996), 268–292CrossRefMATHMathSciNetGoogle Scholar
  42. Fernández-Baca, D., and Lagergren, J. [1998]: On the approximability of the Steiner tree problem in phylogeny. Discrete Applied Mathematics 88 (1998), 129–145CrossRefMATHMathSciNetGoogle Scholar
  43. Fulkerson, D.R. [1972]: Anti-blocking polyhedra. Journal of Combinatorial Theory B 12 (1972), 50–71CrossRefMATHMathSciNetGoogle Scholar
  44. Fürer, M., and Raghavachari, B. [1994]: Approximating the minimum-degree Steiner tree to within one of optimal. Journal of Algorithms 17 (1994), 409–423CrossRefMathSciNetGoogle Scholar
  45. Garey, M.R., and Johnson, D.S. [1976]: The complexity of near-optimal graph coloring. Journal of the ACM 23 (1976), 43–49CrossRefMATHMathSciNetGoogle Scholar
  46. Garey, M.R., Johnson, D.S., and Stockmeyer, L. [1976]: Some simplified NP-complete graph problems. Theoretical Computer Science 1 (1976), 237–267CrossRefMATHMathSciNetGoogle Scholar
  47. Goemans, M.X., and Williamson, D.P. [1994]: New 3/4-approximation algorithms for the maximum satisfiability problem. SIAM Journal on Discrete Mathematics 7 (1994), 656–666CrossRefMATHMathSciNetGoogle Scholar
  48. Goemans, M.X., and Williamson, D.P. [1995]: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42 (1995), 1115–1145CrossRefMATHMathSciNetGoogle Scholar
  49. Grötschel, M., Lovász, L., and Schrijver, A. [1988]: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin 1988MATHGoogle Scholar
  50. Halldórsson, M.M., and Radhakrishnan, J. [1997]: Greed is good: approximating independent sets in sparse and bounded degree graphs. Algorithmica 18 (1997), 145–163CrossRefMATHMathSciNetGoogle Scholar
  51. Håstad, J. [2001]: Some optimal inapproximability results. Journal of the ACM 48 (2001), 798–859CrossRefMATHMathSciNetGoogle Scholar
  52. Heawood, P.J. [1890]: Map colour theorem. Quarterly Journal of Pure Mathematics 24 (1890), 332–338Google Scholar
  53. Hochbaum, D.S. [1982]: Approximation algorithms for the set covering and vertex cover problems. SIAM Journal on Computing 11 (1982), 555–556CrossRefMATHMathSciNetGoogle Scholar
  54. Hochbaum, D.S., and Shmoys, D.B. [1985]: A best possible heuristic for the k-center problem. Mathematics of Operations Research 10 (1985), 180–184CrossRefMATHMathSciNetGoogle Scholar
  55. Holyer, I. [1981]: The NP-completeness of edge-coloring. SIAM Journal on Computing 10 (1981), 718–720CrossRefMATHMathSciNetGoogle Scholar
  56. Hougardy, S., Prömel, H.J., and Steger, A. [1994]: Probabilistically checkable proofs and their consequences for approximation algorithms. Discrete Mathematics 136 (1994), 175–223CrossRefMATHMathSciNetGoogle Scholar
  57. Hsu, W.L., and Nemhauser, G.L. [1979]: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1 (1979), 209–216CrossRefMATHMathSciNetGoogle Scholar
  58. Johnson, D.S. [1974]: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9 (1974), 256–278CrossRefMATHMathSciNetGoogle Scholar
  59. Khanna, S., Linial, N., and Safra, S. [2000]: On the hardness of approximating the chromatic number. Combinatorica 20 (2000), 393–415CrossRefMATHMathSciNetGoogle Scholar
  60. Khot, S., and Regev, O. [2008]: Vertex cover might be hard to approximate to within 2 − ε. Journal of Computer and System Sciences 74 (2008), 335–349CrossRefMATHMathSciNetGoogle Scholar
  61. Knuth, D.E. [1969]: The Art of Computer Programming; Vol. 2. Seminumerical Algorithms. Addison-Wesley, Reading 1969 (third edition: 1997)Google Scholar
  62. König, D. [1916]: Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Mathematische Annalen 77 (1916), 453–465CrossRefMATHMathSciNetGoogle Scholar
  63. Lieberherr, K., and Specker, E. [1981]: Complexity of partial satisfaction. Journal of the ACM 28 (1981), 411–421CrossRefMATHMathSciNetGoogle Scholar
  64. Lovász, L. [1972]: Normal hypergraphs and the perfect graph conjecture. Discrete Mathematics 2 (1972), 253–267CrossRefMATHMathSciNetGoogle Scholar
  65. Lovász, L. [1975]: On the ratio of optimal integral and fractional covers. Discrete Mathematics 13 (1975), 383–390CrossRefMATHMathSciNetGoogle Scholar
  66. Lovász, L. [1979a]: On the Shannon capacity of a graph. IEEE Transactions on Information Theory 25 (1979), 1–7CrossRefMATHGoogle Scholar
  67. Lovász, L. [1979b]: Graph theory and integer programming. In: Discrete Optimization I; Annals of Discrete Mathematics 4 (P.L. Hammer, E.L. Johnson, B.H. Korte, eds.), North-Holland, Amsterdam 1979, pp. 141–158Google Scholar
  68. Lovász, L. [2003]: Semidefinite programs and combinatorial optimization. In: Recent Advances in Algorithms and Combinatorics (B.A. Reed, C. Linhares Sales, eds.), Springer, New York 2003, pp. 137–194Google Scholar
  69. Mahajan, S., and Ramesh, H. [1999]: Derandomizing approximation algorithms based on semidefinite programming. SIAM Journal on Computing 28 (1999), 1641–1663CrossRefMATHMathSciNetGoogle Scholar
  70. Papadimitriou, C.H., and Steiglitz, K. [1982]: Combinatorial Optimization; Algorithms and Complexity. Prentice-Hall, Englewood Cliffs 1982, pp. 406–408Google Scholar
  71. Papadimitriou, C.H., and Yannakakis, M. [1991]: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43 (1991), 425–440CrossRefMATHMathSciNetGoogle Scholar
  72. Papadimitriou, C.H., and Yannakakis, M. [1993]: The traveling salesman problem with distances one and two. Mathematics of Operations Research 18 (1993), 1–12CrossRefMATHMathSciNetGoogle Scholar
  73. Raghavan, P., and Thompson, C.D. [1987]: Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7 (1987), 365–374CrossRefMATHMathSciNetGoogle Scholar
  74. Raz, R., and Safra, S. [1997]: A sub constant error probability low degree test, and a sub constant error probability PCP characterization of NP. Proceedings of the 29th Annual ACM Symposium on Theory of Computing (1997), 475–484Google Scholar
  75. Robertson, N., Sanders, D.P., Seymour, P., and Thomas, R. [1996]: Efficiently four-coloring planar graphs. Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (1996), 571–575Google Scholar
  76. Robertson, N., Sanders, D.P., Seymour, P., and Thomas, R. [1997]: The four colour theorem. Journal of Combinatorial Theory B 70 (1997), 2–44CrossRefMATHMathSciNetGoogle Scholar
  77. Sanders, P., and Steurer, D. [2008]: An asymptotic approximation scheme for multigraph edge coloring. ACM Transactions on Algorithms 4 (2008), Article 21Google Scholar
  78. Singh, M. and Lau, L.C. [2007]: Approximating minimum bounded degree spanning trees to within one of optimal. Proceedings of the 39th Annual ACM Symposium on Theory of Computing (2007), 661–670Google Scholar
  79. Slavík, P. [1997]: A tight analysis of the greedy algorithm for set cover. Journal of Algorithms 25 (1997), 237–254CrossRefMATHMathSciNetGoogle Scholar
  80. Stockmeyer, L.J. [1973]: Planar 3-colorability is polynomial complete. ACM SIGACT News 5 (1973), 19–25CrossRefGoogle Scholar
  81. Trevisan, L. [2004]: On local versus global satisfiability. SIAM Journal on Discrete Mathematics 17 (2004), 541–547CrossRefMATHMathSciNetGoogle Scholar
  82. Vizing, V.G. [1964]: On an estimate of the chromatic class of a p-graph. Diskret. Analiz 3 (1964), 23–30 [in Russian]Google Scholar
  83. Wigderson, A. [1983]: Improving the performance guarantee for approximate graph coloring. Journal of the ACM 30 (1983), 729–735CrossRefMATHMathSciNetGoogle Scholar
  84. Yannakakis, M. [1994]: On the approximation of maximum satisfiability. Journal of Algorithms 17 (1994), 475–502CrossRefMATHMathSciNetGoogle Scholar
  85. Zuckerman, D. [2007]: Linear degree extractors and the inapproximability of Max Clique and Chromatic Number. Theory of Computing 3 (2007), 103–128CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Research Institute for Discrete MathematicsUniversity of BonnBonnGermany

Personalised recommendations