Advertisement

Abstract

We study the approximability of a number of graph problems: treewidth and pathwidth of graphs, one-shot black (and black-white) pebbling costs of directed acyclic graphs, and a variety of different graph layout problems such as minimum cut linear arrangement and interval graph completion. We show that, assuming the recently introduced Small Set Expansion Conjecture, all of these problems are hard to approximate within any constant factor.

Keywords

Directed Acyclic Graph Interval Graph Tree Decomposition Layout Problem Path Decomposition 
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. 1.
    Ambuhl, C., Mastrolilli, M., Svensson, O.: Inapproximability Results for Sparsest Cut, Optimal Linear Arrangement, and Precedence Constrained Scheduling. In: Proceedings of the IEEE Symposium on Foundations of Computer Science, pp. 329–337 (2007)Google Scholar
  2. 2.
    Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Algebraic Discrete Methods 8, 277–284 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Arora, S., Barak, B., Steurer, D.: Subexponential Algorithms for Unique Games and Related Problems. In: FOCS, pp. 563–572 (2010)Google Scholar
  4. 4.
    Austrin, P., Pitassi, T., Wu, Y.: Inapproximability of treewidth, one-shot pebbling, and related layout problems. CoRR, abs/1109.4910 (2011)Google Scholar
  5. 5.
    Barak, B., Raghavendra, P., Steurer, D.: Rounding Semidefinite Programming Hierarchies via Global Correlation. In: FOCS, pp. 472–481 (2011)Google Scholar
  6. 6.
    Bodlaender, H.L.: Treewidth: Structure and Algorithms. In: Prencipe, G., Zaks, S. (eds.) SIROCCO 2007. LNCS, vol. 4474, pp. 11–25. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Bodlaender, H.L.: A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bodlaender, H.L.: Discovering Treewidth. In: Vojtáš, P., Bieliková, M., Charron-Bost, B., Sýkora, O. (eds.) SOFSEM 2005. LNCS, vol. 3381, pp. 1–16. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Bodlaender, H.L., Gilbert, J.R., Hafsteinsson, H., Kloks, T.: Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. Journal of Algorithms 18(2), 238–255 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Charikar, M., Hajiaghayi, M., Karloff, H., Rao, S.: \(l_2^2\) spreading metrics for vertex ordering problems. Algorithmica 56, 577–604 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Courcelle, B.: Graph Rewriting: An Algebraic and Logic Approach. In: Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics (B), pp. 193–242 (1990)Google Scholar
  12. 12.
    Dubey, C.K., Feige, U., Unger, W.: Hardness results for approximating the bandwidth. J. Comput. Syst. Sci. 77(1), 62–90 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Feige, U., Hajiaghayi, M., Lee, J.R.: Improved approximation algorithms for minimum-weight vertex separators. In: Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing, pp. 563–572 (2005)Google Scholar
  14. 14.
    Guruswami, V., Sinop, A.K.: Lasserre Hierarchy, Higher Eigenvalues, and Approximation Schemes for Graph Partitioning and Quadratic Integer Programming with PSD Objectives. In: FOCS, pp. 482–491 (2011)Google Scholar
  15. 15.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of the ACM Symposium on Theory of Computing, STOC 2002, pp. 767–775 (2002)Google Scholar
  16. 16.
    Kinnersley, N.G.: The vertex separation number of a graph equals its path-width. Information Processing Letters 42(6), 345–350 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kirousis, L.M., Papadimitriou, C.H.: Searching and pebbling. Theor. Comput. Sci. 47, 205–218 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Lengauer, T.: Black-white pebbles and graph separation. Acta Informatica 16, 465–475 (1981), doi:10.1007/BF00264496MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Lengauer, T., Tarjan, R.E.: Asymptotically tight bounds on time-space trade-offs in a pebble game. J. ACM 29, 1087–1130 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Nordström, J.: New wine into old wineskins: A survey of some pebbling classics with supplemental results. Draft manuscript (November 2010)Google Scholar
  21. 21.
    Raghavendra, P., Steurer, D.: Graph expansion and the unique games conjecture. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, pp. 755–764. ACM, New York (2010)Google Scholar
  22. 22.
    Raghavendra, P., Steurer, D., Tulsiani, M.: Reductions Between Expansion Problems. To appear in CCC (2012)Google Scholar
  23. 23.
    Ramalingam, G., Rangan, C.P.: A unified approach to domination problems on interval graphs. Inf. Process. Lett. 27, 271–274 (1988)zbMATHCrossRefGoogle Scholar
  24. 24.
    Robertson, N., Seymour, P.D.: Graph minors. III. Planar tree-width. J. Comb. Theory, Ser. B 36(1), 49–64 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. Journal of Algorithms 7(3), 309–322 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Sethi, R.: Complete register allocation problems. In: Proceedings of the Fifth Annual ACM Symposium on Theory of Computing, STOC 1973, pp. 182–195 (1973)Google Scholar
  27. 27.
    Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Per Austrin
    • 1
  • Toniann Pitassi
    • 1
  • Yu Wu
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoCanada

Personalised recommendations