Advertisement

Some Recent Progress for Approximation Algorithms

  • Ken-ichi Kawarabayashi
Part of the Lecture Notes in Physics book series (LNP, volume 911)

Abstract

We survey some recent progress on approximation algorithms. Our main focus is the following two problems that have some recent breakthroughs; the edge-disjoint paths problem and the graph coloring problem. These breakthroughs involve the following three ingredients that are quite central in approximation algorithms: (1) Combinatorial (graph theoretical) approach, (2) LP based approach and (3) Semi-definite programming approach. We also sketch how they are used to obtain recent development.

Keywords

Approximation algorithm Hardness 

References

  1. 1.
    M. Andrews, J. Chuzhoy, S. Khanna, L. Zhang, Hardness of the undirected edge-disjoint paths problem with congestion, in Proceedings of the 46th IEEE Symposium on Foundations of Computer Science (FOCS), Pittsburgh (2005), pp. 226–244Google Scholar
  2. 2.
    S. Arora, E. Chlamtac, M. Charikar, New approximation guarantee for chromatic number, in Proceedings of the 38th STOC, Seattle (2006) pp. 215–224Google Scholar
  3. 3.
    S. Arora, S. Rao, U. Vazirani, Expanders, geometric embeddings and graph partitioning. J. ACM 56(2), 1–37 (2009). Announced at STOC’04Google Scholar
  4. 4.
    B. Berger, J. Rompel, A better performance guarantee for approximate graph coloring. Algorithmica, 5(3), 459–466 (1990)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Blum, New approximation algorithms for graph coloring. J. ACM 41(3), 470–516 (1994). Announced at STOC’89 and FOCS’90Google Scholar
  6. 6.
    A. Blum, D. Karger, An \(\tilde{O}(n^{3/14})\)-coloring algorithm for 3-colorable graphs. Inf. Process. Lett. 61(1), 49–53 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    C. Chekuri, S. Khanna, B. Shepherd, The all-or-nothing multicommodity flow problem, in Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), Chicago (2004), pp. 156–165Google Scholar
  8. 8.
    C. Chekuri, S. Khanna, B. Shepherd, An \(O(\sqrt{n})\) approximation and integrality gap for disjoint paths and unsplittable flow. Theory Comput. 2, 137–146 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    C. Chekuri, S. Khanna, B. Shepherd, Edge-disjoint paths in planar graphs with constant congestion. SIAM J. Comput. 39, 281–301 (2009)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    E. Chlamtac, Approximation algorithms using hierarchies of semidefinite programming relaxations, in Proceedings of the 48th FOCS, Providence (2007), pp. 691–701Google Scholar
  11. 11.
    J. Chuzhoy, S. Li, A polylogarithimic approximation algorithm for edge-disjoint paths with congestion 2, in Proceedings of the 53rd IEEE Symposium on Foundations of Computer Science (FOCS), New Brunswick (2012), pp. 233–242Google Scholar
  12. 12.
    I. Dinur, E. Mossel, O. Regev, Conditional hardness for approximate coloring. SIAM J. Comput. 39(3), 843–873 (2009). Announced at STOC’06Google Scholar
  13. 13.
    S. Even, A. Itai, A. Shamir, On the complexity of timetable and multicommodity flow problems. SIAM J. Comput. 5, 691–703 (1976)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    U. Feige, J. Kilian, Zero-knowledge and the chromatic number. J. Comput. Syst. Sci. 57, 187–199 (1998)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    U. Feige, M. Langberg, G. Schechtman, Graphs with tiny vector chromatic numbers and huge chromatic numbers, In Proceedings of the 43rd FOCS, Vancouver (2002), pp. 283–292Google Scholar
  16. 16.
    A. Frank, Packing paths, cuts and circuits – a survey, in Paths, Flows and VLSI-Layout, ed. by B. Korte, L. Lovász, H.J. Promel, A. Schrijver (Springer, Berlin, 1990), pp. 49–100Google Scholar
  17. 17.
    M. Garey, D. Johnson, L. Stockmeyer, Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976). Announced at STOC’74Google Scholar
  18. 18.
    N. Garg, V. Vazirani, M. Yannakakis, Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18, 3–20 (1997)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    M. Goemansl, D. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995). Announced at STOC’94Google Scholar
  20. 20.
    V. Guruswami, S. Khanna, On the hardness of 4-coloring a 3-colorable graph. J. Discret. Math. 18(1), 30–40 (2004)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    V. Guruswami, S. Khanna, R. Rajaraman, B. Shepherd, M. Yannakakis, Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems. J. Comput. Syst. Sci. 67, 473–496 (2003)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    J. Håstad, Clique is hard to approximate within \(n^{1-\varepsilon }\). Acta Math. 182, 105–142 (1999)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    D. Karger, R. Motwani, M. Sudan, Approximate graph coloring by semidefinite programming. J. ACM 45(2), 246–265 (1998). Announced at FOCS’94Google Scholar
  24. 24.
    R.M. Karp, On the computational complexity of combinatorial problems. Networks 5, 45–68 (1975)MATHMathSciNetGoogle Scholar
  25. 25.
    K. Kawarabayashi, Y. Kobayashi, An improved algorithm for the half-disjoint paths problem. SIAM J. Discret. Math. 25, 1322–1330 (2011)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    K. Kawarabayashi, Y. Kobayashi, All-or-nothing multicommodity flow problem with bounded fractionality in planar graphs, in The 54th Annual Symposium on Foundations of Computer Science (FOCS 2013), Berkeley (2013), pp.187–196Google Scholar
  27. 27.
    K. Kawarabayashi, B. Reed, A nearly linear time algorithm for the half integral disjoint paths packing, in Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), San Francisco (2008), pp. 446–454Google Scholar
  28. 28.
    K. Kawarabayashi, M. Thorup, Combinatorial coloring of 3-colorable graphs, in Proceedings of the 53rd FOCS, New Brunswick (2012)Google Scholar
  29. 29.
    K. Kawarabayashi, M. Thorup, Coloring 3-colorable graphs with o(n 1∕5) colors, in The 31st Symposium on Theoretical Aspects of Computer Science (STACS’14), Lyon (2014), pp. 458–469Google Scholar
  30. 30.
    K. Kawarabayashi, Y. Kobayashi, B. Reed, The disjoint paths problem in quadratic time. J. Combin. Theory Ser. B 102, 424–435 (2012)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    S. Khanna, N. Linial, S. Safra, On the hardness of approximating the chromatic number. Combinatorica 20(3), 393–415 (2000)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    J. Kleinberg, Decision algorithms for unsplittable flow and the half-disjoint paths problem, in Proceedings of the 30th ACM Symposium on Theory of Computing (STOC), Dallas (1998), pp. 530–539Google Scholar
  33. 33.
    M.R. Kramer, J. van Leeuwen, The complexity of wire-routing and finding minimum area layouts for arbitrary VLSI circuits. Adv. Comput. Res. 2, 129–146 (1984)Google Scholar
  34. 34.
    N. Robertson, P.D. Seymour, Graph minors. XIII. The disjoint paths problem. J. Combin. Theory Ser. B 63, 65–110 (1995)MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    A. Schrijver, A comparison of the Delsarte and Lovász bounds. IEEE Trans. Inf. Theory 25, 425–429 (1979)MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    A. Schrijver, in Combinatorial Optimization: Polyhedra and Efficiency. Algorithm and Combinatorics, vol. 24 (Springer, Berlin/New York, 2003)Google Scholar
  37. 37.
    L. Séguin-Charbonneau, B.F. Shepherd, Maximum edge-disjoint paths in planar graphs with congestion 2, in Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science (FOCS), Palm Springs (2011), pp. 200–209Google Scholar
  38. 38.
    M. Szegedy, A note on the \(\theta\) number of Lovász and the generalized Delsarte bound, in Proceedings of the 35th FOCS, Santa Fe (1994), pp. 36–39Google Scholar
  39. 39.
    A. Wigderson, Improving the performance guarantee for approximate graph coloring. J. ACM 30(4), 729–735 (1983). Announced at STOC’82Google Scholar

Copyright information

© Springer Japan 2016

Authors and Affiliations

  1. 1.National Institute of InformaticsJST ERATO Kawarabayashi ProjectTokyoJapan

Personalised recommendations