International Workshop on Approximation and Online Algorithms

Approximation and Online Algorithms pp 122-132 | Cite as

An \(O(\log \mathrm{OPT})\)-Approximation for Covering/Packing Minor Models of \(\theta _{r}\)

  • Dimitris Chatzidimitriou
  • Jean-Florent Raymond
  • Ignasi Sau
  • Dimitrios M. Thilikos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9499)

Abstract

Let \(\mathcal{C}_{H}\) be the class of graphs containing some fixed graph H as a minor. We define \(\mathbf{c}^\mathsf{v}_{H}(G)\) (resp. \(\mathbf{c}^\mathsf{e}_{H}(G)\)) as the minimun number of vertices (resp. edges) whose removal from G produces a graph without any subgraph isomorphic to a graph in \(\mathcal{C}_{H}\). Also \(\mathbf{p}^\mathsf{v}_{H}(G)\) (resp. \(\mathbf{p}^\mathsf{e}_{H}(G)\)) is the the maximum number of vertex- (resp. edge-) disjoint subgraphs of G that are isomorphic to some graph in \(\mathcal{C}_{H}\). We denote by \(\theta _{r}\) the graph with two vertices and r parallel edges between them. When \(H=\theta _{r}\), the parameters \(\mathbf{c}^\mathsf{v/e}_{H}\) and \(\mathbf{p}^\mathsf{v/e}_{H}\) are NP-complete to compute (for sufficiently large r). In this paper we prove a series of combinatorial and algorithmic lemmata that imply that if \(\mathbf{p}^\mathsf{v/e}_{\theta _r}(G)\le k\), then \(\mathbf{c}^\mathsf{v/e}_{\theta _r}(G) = O(k\log k)\). This means that for every r, the class \(\mathcal{C}_{\theta _{r}}\) has the vertex/edge Erdős-Pósa property. Using the combinatorial ideas from our proofs we introduce a unified approach for the design of an \(O(\log \mathrm{OPT})\)-approximation algorithm for \(\mathbf{c}^\mathsf{v}_{\theta _{r}}\), \(\mathbf{p}^\mathsf{v}_{\theta _{r}}\), \(\mathbf{c}^\mathsf{e}_{\theta _{r}}\) and \(\mathbf{p}^\mathsf{e}_{\theta _{r}}\) that runs in \(O(n\cdot \log (n)\cdot m)\) steps.

Keywords

Erdős-Pósa properties Minors Graph packing Covering 

References

  1. 1.
    Caprara, A.P.A., Rizzi, R.: Packing cycles in undirected graphs. J. Algorithms 48(1), 239–256 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bazgan, C., Tuza, Z., Vanderpooten, D.: Efficient algorithms for decomposing graphs under degree constraints. Discrete Appl. Math. 155(8), 979–988 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Birmelé, E., Bondy, J.A., Reed, B.A.: The Erdős-Pósa property for long circuits. Combinatorica 27, 135–145 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) kernelization. In: Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 629–638. IEEE Computer Society, Washington, DC (2009)Google Scholar
  5. 5.
    Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) kernelization. CoRR, abs/0904.0727 (2009)
  6. 6.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theoret. Comput. Sci. 412(35), 4570–4578 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chatzidimitriou, D., Raymond, J-F., Sau, I., Thilikos, D.M.: Minors in graphs of large \(\theta _r\)-girth. CoRR, abs/1510.03041 (2015)
  8. 8.
    Chatzidimitriou, D., Raymond, J-F., Sau, I., Thilikos, D.M.: An \(o(\log {OPT})\)-approximation for covering and packing minor models of the pumpkin. CoRR, abs/1510.03945 (2015)
  9. 9.
    Chekuri, C., Chuzhoy, J.: Large-treewidth graph decompositions and applications. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing (STOC), pp. 291–300. ACM, New York (2013)Google Scholar
  10. 10.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, 3rd edn. Springer, Heidelberg (2005)MATHGoogle Scholar
  11. 11.
    Diestel, R., Kawarabayashi, K.I., Wollan, P.: The Erdős-Pósa property for clique minors in highly connected graphs. J. Comb. Theor. Series B 102(2), 454–469 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Erdős, P., Pósa, L.: On independent circuits contained in a graph. Can. J. Math. 17, 347–352 (1965)CrossRefGoogle Scholar
  13. 13.
    Fiorini, S., Joret, G., Pietropaoli, U.: Hitting diamonds and growing cacti. In: Eisenbrand, F., Shepherd, F.B. (eds.) IPCO 2010. LNCS, vol. 6080, pp. 191–204. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Fiorini, S., Joret, G., Sau, I.: Optimal Erdős-Pósa property for pumpkins. Manuscript (2013)Google Scholar
  15. 15.
    Fiorini, S., Joret, G., Wood, D.R.: Excluded forest minors and the erdős-pósa property. Comb. Probab. Comput. 22(5), 700–721 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fomin, F.V., Lokshtanov, D., Misra, N., Saurabh, S.: Planar F-deletion: approximation, kernelization and optimal FPT-algorithms. In: IEEE 53rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 470–479, October 2012Google Scholar
  17. 17.
    Jim, G., Kasper, K.: The Erdős-Pósa property for matroid circuits. J. Comb. Theor. Ser. B 99(2), 407–419 (2009)CrossRefMATHGoogle Scholar
  18. 18.
    Golovach, P.A.: Personal communication (2015)Google Scholar
  19. 19.
    Joret, G., Paul, C., Sau, I., Saurabh, S., Thomassé, S.: Hitting and harvesting pumpkins. SIAM J. Discr. Math. 28(3), 1363–1390 (2014)CrossRefMATHGoogle Scholar
  20. 20.
    Kakimura, N., Kawarabayashi, K.I., Kobayashi, Y.: Erdős-pósa property and its algorithmic applications: Parity constraints, subset feedback set, and subset packing. In: Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1726–1736. SIAM (2012)Google Scholar
  21. 21.
    Kawarabayashi, K.I., Kobayashi, Y.: Edge-disjoint odd cycles in 4-edge-connected graphs. In: 29th International Symposium on Theoretical Aspects of Computer Science, (STACS), February 29th – March 3rd, 2012, Paris, France, pp. 206–217 (2012)Google Scholar
  22. 22.
    Kawarabayashi, K.-I., Nakamoto, A.: The erdős-pósa property for vertex- and edge-disjoint odd cycles in graphs on orientable surfaces. Discr. Math. 307(6), 764–768 (2007)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kloks, T., Lee, C.M., Liu, J.: New algorithms for \(k\)-face cover, \(k\)-feedback vertex set, and \(k\)-disjoint cycles on plane and planar graphs. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 282–295. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  24. 24.
    Král’, D., Voss, H.-J.: Edge-disjoint odd cycles in planar graphs. J. Comb. Theor. Ser. B 90(1), 107–120 (2004)CrossRefMATHGoogle Scholar
  25. 25.
    Krivelevich, M., Nutov, Z., Salavatipour, M.R., Yuster, J.V., Yuster, R.: Approximation algorithms and hardness results for cycle packing problems. ACM Trans. Algorithms 3(4) (2007)Google Scholar
  26. 26.
    Krivelevich, M., Nutov, Z., Yuster, R.: Approximation algorithms for cycle packing problems. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 556–561. SIAM, Philadelphia, PA (2005)Google Scholar
  27. 27.
    Pontecorvi, M., Wollan, P.: Disjoint cycles intersecting a set of vertices. J. Comb. Theor. Ser. B 102(5), 1134–1141 (2012)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Rautenbach, D., Reed, B.: The Erdős-Pósa property for odd cycles in highly connected graphs. Combinatorica 21, 267–278 (2001)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Raymond, J.-F., Sau, I., Thilikos, D.M.: An edge variant of the Erdős-pósa property. CoRR, abs/1311.1108 (2013)
  30. 30.
    Reed, B.A., Robertson, N., Seymour, P.D., Thomas, R.: Packing directed circuits. Combinatorica 16(4), 535–554 (1996)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Robertson, N., Seymour, P.D.: Graph minors. v. excluding a planar graph. J. Comb. Theor. Ser. B 41(2), 92–114 (1986)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Salavatipour, M.R., Verstraete, J.: Disjoint cycles: integrality gap, hardness, and approximation. In: Jünger, M., Kaibel, V. (eds.) IPCO 2005. LNCS, vol. 3509, pp. 51–65. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Dimitris Chatzidimitriou
    • 1
  • Jean-Florent Raymond
    • 2
    • 3
  • Ignasi Sau
    • 2
  • Dimitrios M. Thilikos
    • 1
    • 2
  1. 1.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece
  2. 2.AlGCo Project-TeamCNRS, LIRMMMontpellierFrance
  3. 3.Computer Science InstituteUniversity of WarsawWarsawPoland

Personalised recommendations