Approximation algorithms via contraction decomposition

Abstract

We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge deletions instead of contractions, obtained in [4,20, 10, 17], and it generalizes a similar result for “compression” (a variant of contraction) in planar graphs [29]. Our decomposition result is a powerful tool for obtaining PTASs for contraction-closed problems (whose optimal solution only improves under contraction), a much more general class than minor-closed problems. We prove that any contraction-closed problem satisfying just a few simple conditions has a PTAS in bounded-genus graphs. In particular, our framework yields PTASs for the weighted Traveling Salesman Problem and for minimum-weight c-edge-connected submultigraph on bounded-genus graphs, improving and generalizing previous algorithms of [24, 1, 29, 25, 8, 5]. We also highlight the only main difficulty in extending our results to general H-minor-free graphs.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    Sanjeev Arora, Michelangelo Grigni, David Karger, Philip Klein and Andrzej Woloszyn: A polynomial-time approximation scheme for weighted planar graph TSP, in: Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 33–41, 1998.

  2. [2]

    Eyal Amir: Efficient approximation for triangulation of minimum treewidth, in: Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence (UAI-2001), pages 7–15, Morgan Kaufmann Publishers, San Francisco, CA, 2001.

    Google Scholar 

  3. [3]

    Sanjeev Arora, Satish Rao and Umesh Vazirani: Expander flows, geometric embeddings and graph partitioning; in: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pages 222–231, 2004.

  4. [4]

    Brenda S. Baker: Approximation algorithms for NP-complete problems on planar graphs, Journal of the Association for Computing Machinery41(1) (1994), 153–180.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    André Berger, Artur Czumaj, Michelangelo Grigni and Hairong Zhao: Approximation schemes for minimum 2-connected spanning subgraphs in weighted planar graphs, in: Proceedings of the 13th Annual European Symposium on Algorithms, volume 3669 of Lecture Notes in Computer Science, pages 472–483, Palma de Mallorca, Spain, October 2005.

    Google Scholar 

  6. [6]

    Richard Brunet, Bojan Mohar and R. Bruce Richter: Separating and non-separating disjoint homotopic cycles in graph embeddings, Journal of Combinatorial Theory, Series B66(2) (1996), 201–231.

    MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    Hans L. Bodlaender: Discovering treewidth, in: Proceedings of the 31st Conference on Current Trends in Theory and Practice of Computer Science, volume 3381 of Lecture Notes in Computer Science, pages 1–16, Liptovský Ján, Slovakia, January 2005.

    Google Scholar 

  8. [8]

    Artur Czumaj, Michelangelo Grigni, Papa Sissokho and Hairong Zhao: Approximation schemes for minimum 2-edge-connected and biconnected subgraphs in planar graphs, in: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 496–505, Society for Industrial and Applied Mathematics. Philadelphia, PA, USA, 2004.

    Google Scholar 

  9. [9]

    Sergio Cabello and Bojan Mohar: Finding shortest non-separating and non-contractible cycles for topologically embedded graphs, in: Proceedings of the 13th Annual European Symposium on Algorithms, volume 3669 of Lecture Notes in Computer Science, pages 131–142, Palma de Mallorca, Spain, October 2005.

    Google Scholar 

  10. [10]

    Matt DeVos, Guoli Ding, Bogdan Oporowski, Daniel P. Sanders, Bruce Reed, Paul Seymour and Dirk Vertigan: Excluding any graph as a minor allows a low tree-width 2-coloring, Journal of Combinatorial Theory, Series B91(1) (2004), 25–41.

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi and Dimitrios M. Thilikos: Bidimensional parameters and local treewidth, SIAM Journal on Discrete Mathematics18(3) (2004), 501–511.

    MathSciNet  MATH  Article  Google Scholar 

  12. [12]

    Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi and Dimitrios M. Thilikos: Fixed-parameter algorithms for (k, r)-center in planar graphs and map graphs, ACM Transactions on Algorithms1(1) (2005), 33–47.

    MathSciNet  Article  Google Scholar 

  13. [13]

    Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi and Dimitrios M. Thilikos: Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs, Journal of the ACM52(6) (2005), 866–893.

    MathSciNet  Article  Google Scholar 

  14. [14]

    Frederic Dorn, Fedor V. Fomin and Dimitrios M. Thilikos: Fast subexponential algorithm for non-local problems on graphs of bounded genus, in: Proceedings of the 10th Scandinavian Workshop on Algorithm Theory, volume 4059 of Lecture Notes in Computer Science, pages 172–183, Riga, Latvia, July 2006.

  15. [15]

    Erik D. Demaine and MohammadTaghi Hajiaghayi: Diameter and treewidth in minor-closed graph families, revisited; Algorithmica40(3) (2004), 211–215.

    MathSciNet  MATH  Article  Google Scholar 

  16. [16]

    Erik D. Demaine and MohammadTaghi Hajiaghayi: Equivalence of local treewidth and linear local treewidth and its algorithmic applications, in: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA’04), pages 833–842, January 2004.

  17. [17]

    Erik D. Demaine, MohammadTaghi Hajiaghayi and Ken-Ichi Kawarabayashi: Algorithmic graph minor theory: Decomposition, approximation, and coloring; in: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pages 637–646, Pittsburgh, PA, October 2005.

  18. [18]

    Erik D. Demaine, MohammadTaghi Hajiaghayi, Naomi Nishimura, Prabhakar Ragde and Dimitrios M. Thilikos: Approximation algorithms for classes of graphs excluding single-crossing graphs as minors, Journal of Computer and System Sciences69(2) (2004), 166–195.

    MathSciNet  MATH  Article  Google Scholar 

  19. [19]

    Erik D. Demaine, MohammadTaghi Hajiaghayi and Dimitrios M. Thilikos: The bidimensional theory of bounded-genus graphs, SIAM Journal on Discrete Mathematics20(2) (2006), 357–371.

    MathSciNet  MATH  Article  Google Scholar 

  20. [20]

    David Eppstein: Diameter and treewidth in minor-closed graph families, Algorithmica27(3–4) (2000), 275–291.

    MathSciNet  MATH  Article  Google Scholar 

  21. [21]

    Uriel Feige and Joe Kilian: Zero knowledge and the chromatic number, Journal of Computer and System Sciences57(2) (1998), 187–199.

    MathSciNet  MATH  Article  Google Scholar 

  22. [22]

    Fedor V. Fomin and Dimitrios M. Thilikos: Fast parameterized algorithms for graphs on surfaces: Linear kernel and exponential speed-up, in: Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004), pages 581–592, Turku, Finland, July 2004.

  23. [23]

    John R. Gilbert, Joan P. Hutchinson and Robert Endre Tarjan: A separator theorem for graphs of bounded genus, J. Algorithms5(3) (1984), 391–407.

    MathSciNet  MATH  Article  Google Scholar 

  24. [24]

    Michelangelo Grigni, Elias Koutsoupias and Christos Papadimitriou: An approximation scheme for planar graph TSP, in: Proceedings of the 36th Annual Symposium on Foundations of Computer Science (Milwaukee, WI, 1995), pages 640–645, Los Alamitos, CA, 1995.

  25. [25]

    Michelangelo Grigni: Approximate TSP in graphs with forbidden minors, in: Proceedings of the 27th International Colloquium of Automata, Languages and Programming (Geneva, 2000), volume 1853 of Lecture Notes in Computer Science, pages 869–877. Springer, Berlin, 2000.

    Google Scholar 

  26. [26]

    Martin Grohe: Local tree-width, excluded minors, and approximation algorithms; Combinatorica23(4) (2003), 613–632.

    MathSciNet  MATH  Article  Google Scholar 

  27. [27]

    Michelangelo Grigni and Papa Sissokho: Light spanners and approximate TSP in weighted graphs with forbidden minors, in: Proceedings of the 13th Annual ACMSIAM Symposium on Discrete Algorithms, pages 852–857, 2002.

  28. [28]

    Jonathan A. Kelner: Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus; SIAM Journal on Computing35(4) (2006), 882–902 (electronic).

    MathSciNet  MATH  Article  Google Scholar 

  29. [29]

    Philip N. Klein: A linear-time approximation scheme for TSP for planar weighted graphs, in: Proceedings of the 46th IEEE Symposium on Foundations of Computer Science, pages 146–155, 2005.

  30. [30]

    Philip N. Klein: A subset spanner for planar graphs, with application to subset TSP, in: Proceedings of the 38th ACM Symposium on Theory of Computing, pages 749–756, Seattle, WA, 2006.

  31. [31]

    Tom Leighton and Satish Rao: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms, Journal of the ACM46(6) (1999), 787–832.

    MathSciNet  MATH  Article  Google Scholar 

  32. [32]

    Richard J. Lipton and Robert Endre Tarjan: Applications of a planar separator theorem, SIAM Journal on Computing9(3) (1980), 615–627.

    MathSciNet  MATH  Article  Google Scholar 

  33. [33]

    Bojan Mohar: Combinatorial local planarity and the width of graph embeddings, Canadian Journal of Mathematics44(6) (1992), 1272–1288.

    MathSciNet  MATH  Article  Google Scholar 

  34. [34]

    Bojan Mohar: A linear time algorithm for embedding graphs in an arbitrary surface, SIAM Journal on Discrete Mathematics12(1) (1999), 6–26.

    MathSciNet  MATH  Article  Google Scholar 

  35. [35]

    Bojan Mohar: Graph minors and graphs on surfaces, in: Surveys in Combinatorics, volume 288 of London Math. Soc. Lecture Note Ser., pages 145–163. Cambridge Univ. Press, Cambridge, 2001.

    Google Scholar 

  36. [36]

    Bojan Mohar and Carsten Thomassen: Graphs on surfaces, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2001.

    Google Scholar 

  37. [37]

    Neil Robertson and Paul D. Seymour: Graph minors II.: Algorithmic aspects of tree-width; Journal of Algorithms7(3) (1986), 309–322.

    MathSciNet  MATH  Article  Google Scholar 

  38. [38]

    Neil Robertson and Paul D. Seymour: Graph minors XI.: Circuits on a surface; Journal of Combinatorial Theory, Series B60(1) (1994), 72–106.

    MathSciNet  MATH  Article  Google Scholar 

  39. [39]

    Neil Robertson and Paul D. Seymour: Graph minors XVI.: Excluding a non-planar graph; Journal of Combinatorial Theory, Series B89(1) (2003), 43–76.

    MathSciNet  MATH  Article  Google Scholar 

  40. [40]

    Robin Thomas: Problem Session of the 3rd Solvene Conference on Graph Theory, Bled, Slovenia, 1995.

  41. [41]

    Mikkel Thorup: All structured programs have small tree-width and good register allocation, Information and Computation142(2) (1998), 159–181.

    MathSciNet  MATH  Article  Google Scholar 

  42. [42]

    Klaus Wagner: Über eine Eigenschaft der ebenen Komplexe, Mathematische Annalen114 (1937), 570–590.

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Erik D. Demaine.

Additional information

A preliminary version of this paper appeared in Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, January 2007.

on leave from: Department of Mathematics University of Ljubljana 1000 Ljubljana Slovenia

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Demaine, E.D., Hajiaghayi, M. & Mohar, B. Approximation algorithms via contraction decomposition. Combinatorica 30, 533–552 (2010). https://doi.org/10.1007/s00493-010-2341-5

Download citation

Mathematics Subject Classification (2000)

  • 05C83
  • 05C10
  • 68R10