, Volume 80, Issue 8, pp 2221–2239 | Cite as

Approximability of Clique Transversal in Perfect Graphs

  • Samuel Fiorini
  • R. Krithika
  • N. S. Narayanaswamy
  • Venkatesh Raman


Given an undirected simple graph G, a set of vertices is an r-clique transversal if it has at least one vertex from every r-clique. Such sets generalize vertex covers as a vertex cover is a 2-clique transversal. Perfect graphs are a well-studied class of graphs on which a minimum weight vertex cover can be obtained in polynomial time. Further, an r-clique transversal in a perfect graph is also a set of vertices whose deletion results in an \((r-1)\)-colorable graph. In this work, we study the problem of finding a minimum weight r-clique transversal in a perfect graph. This problem is known to be \(\mathsf {NP}\)-hard for \(r \ge 3\) and admits a straightforward r-approximation algorithm. We describe two different \(\frac{r+1}{2}\)-approximation algorithms for the problem. Both the algorithms are based on (different) linear programming relaxations. The first algorithm employs the primal–dual method while the second uses rounding based on a threshold value. We also show that the problem is APX-hard and describe hardness results in the context of parameterized algorithms and kernelization.


r-Clique transversal Odd cycle transversal Perfect graphs Approximation algorithms Linear programming 


  1. 1.
    Abu-Khzam, F.N.: A kernelization algorithm for \(d\)-hitting set. J. Comput. Syst. Sci. 76(7), 524–531 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abu-Khzam, F.N., Fernau, H.: Kernels: annotated, proper and induced. In: Parameterized and Exact Computation, Volume 4169 of Lecture Notes in Computer Science, pp. 264–275. Springer, Berlin (2006)Google Scholar
  3. 3.
    Berge, C.: Färbung von graphen, deren sämtliche bzw. deren ungerade kreise starr sind (zusammenfassung). Wissenschaftliche Zeitschrift, Martin Luther Universität Halle-Wittenberg Mathematisch-Naturwissenschaftliche Reihe 10, 114–115 (1961)Google Scholar
  4. 4.
    Berry, L.A., Kennedy, W.S., King, A.D., Li, Z., Reed, B.A.: Finding a maximum-weight induced \(k\)-partite subgraph of an \(i\)-triangulated graph. Discret. Appl. Math. 158(7), 765–770 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chudnovsky, M., Cornuéjols, G., Liu, X., Seymour, P.D., Vusković, K.: Recognizing berge graphs. Combinatorica 25(2), 143–186 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cygan, M., Fomin, F .V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. J. Comput. Syst. Sci. 67, 789–807 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chudnovsky, M., Robertson, N., Seymour, P.D., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164(1), 51–229 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162(1), 439–485 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM 61(4), 23:1–23:27 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Guruswami, V., Lee, E.: Inapproximability of feedback vertex set for bounded length cycles. Electron. Colloq. Comput. Complex. 21, 6 (2014)Google Scholar
  12. 12.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  13. 13.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, 2nd edn. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  14. 14.
    Hochbaum, D .S.: Approximation Algorithms for NP-Hard Problems. PWS Publishing Co, Boston (1997)zbMATHGoogle Scholar
  15. 15.
    Iwata, Y., Oka, K., Yuichi, Y.: Linear-time FPT algorithms via network flow. In: Proceedings of the ACM–SIAM Symposium on Discrete Algorithms, pp. 1749–1761 (2014)Google Scholar
  16. 16.
    Impagliazzo, R., Paturi, R.: Complexity of k-SAT. In: Proceedings of IEEE Conference on Computational Complexity, pp. 237–240 (1999)Google Scholar
  17. 17.
    Iwata, Y., Wahlström, M., Yoshida, Y.: Half-integrality, LP-branching, and FPT algorithms. SIAM J. Comput. 45(4), 1377–1411 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Krithika, R., Narayanaswamy, N.S.: Another disjoint compression algorithm for odd cycle transversal. Inf. Process. Lett. 113(22–24), 849–851 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Krithika, R., Narayanaswamy, N.S.: Parameterized algorithms for \((r, l)\)-partization. J. Graph Algorithms Appl. 17(2), 129–146 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Knuth, D.E.: The sandwich theorem. Electron. J. Comb. 1(A1), 1–48 (1994)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within \(2-\epsilon \). J. Comput. Syst. Sci. 74(3), 335–349 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lokshtanov, D., Narayanaswamy, N .S., Raman, V., Ramanujan, M .S., Saurabh, S.: Faster parameterized algorithms using linear programming. ACM Trans. Algorithms 11(2), 15:1–15:31 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lovász, L.: Normal hypergraphs and the perfect graph conjecture. Discret. Math. 2(3), 253–267 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lovász, L.: On Minimax Theorems of Combinatorics. Ph.D thesis, Matemathikai Lapok, vol. 26, pp. 209–264 (1975)Google Scholar
  25. 25.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20(2), 219–230 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nemhauser, G.L., Trotter, L.E.: Properties of vertex packing and independence system polyhedra. Math. Program. 6(1), 48–61 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nemhauser, G.L., Trotter, L.E.: Vertex packings: structural properties and algorithms. Math. Program. 8(1), 232–248 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32, 299–301 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wagler, A.: Critical and anticritical edges in perfect graphs. In: Graph-Theoretic Concepts in Computer Science, Volume 2204 of Lecture Notes in Computer Science, pp. 317–327 (2001)Google Scholar
  30. 30.
    Williamson, D .P., Shmoys, D .B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Samuel Fiorini
    • 1
  • R. Krithika
    • 2
  • N. S. Narayanaswamy
    • 3
  • Venkatesh Raman
    • 2
  1. 1.Département de MathématiqueUniversité libre de BruxellesBrusselsBelgium
  2. 2.The Institute of Mathematical SciencesHBNIChennaiIndia
  3. 3.Department of Computer Science and EngineeringIndian Institute of Technology MadrasChennaiIndia

Personalised recommendations