Upper Domination: Complexity and Approximation

  • Cristina BazganEmail author
  • Ljiljana Brankovic
  • Katrin Casel
  • Henning Fernau
  • Klaus Jansen
  • Kim-Manuel Klein
  • Michael Lampis
  • Mathieu Liedloff
  • Jérôme Monnot
  • Vangelis Th. Paschos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)


We consider Upper Domination, the problem of finding a maximum cardinality minimal dominating set in a graph. We show that this problem does not admit an \(n^{1-\epsilon }\) approximation for any \(\epsilon >0\), making it significantly harder than Dominating Set, while it remains hard even on severely restricted special cases, such as cubic graphs (APX-hard), and planar subcubic graphs (NP-hard). We complement our negative results by showing that the problem admits an \(O(\varDelta )\) approximation on graphs of maximum degree \(\varDelta \), as well as an EPTAS on planar graphs. Along the way, we also derive essentially tight \(n^{1-\frac{1}{d}}\) upper and lower bounds on the approximability of the related problem Maximum Minimal Hitting Set on d-uniform hypergraphs, generalising known results for Maximum Minimal Vertex Cover.



We thank anonymous referees for helpful comments. We also thank our colleagues David Manlove and Daniel Meister for discussions on upper domination. Part of this research was supported by the DFG, grant FE 560/6-1.


  1. 1.
    AbouEisha, H., Hussain, S., Lozin, V., Monnot, J., Ries, B.: A dichotomy for upper domination in monogenic classes. In: Zhang, Z., Wu, L., Xu, W., Du, D.-Z. (eds.) COCOA 2014. LNCS, vol. 8881, pp. 258–267. Springer, Heidelberg (2014)Google Scholar
  2. 2.
    Arkin, E.M., Bender, M.A., Mitchell, J.S.B., Skiena, S.: The lazy bureaucrat scheduling problem. Inf. Comput. 184(1), 129–146 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 153–180 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bender, M.A., Clifford, R., Tsichlas, K.: Scheduling algorithms for procrastinators. J. Sched. 11(2), 95–104 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berman, P., Fujito, T.: On the approximation properties of independent set problem in degree 3 graphs. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 449–460. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  6. 6.
    Bodlaender, H.L.: A partial \(k\)-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209, 1–45 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boria, N., Della Croce, F.D., Paschos, V.T.: On the max min vertex cover Problem. In: Kaklamanis, C., Pruhs, K. (eds.) WAOA 2013. LNCS, vol. 8447, pp. 37–48. Springer, Heidelberg (2014)Google Scholar
  8. 8.
    Boros, E., Gurvich, V., Hammer, P.L.: Dual subimplicants of positive boolean functions. Optim. Methods Softw. 10(2), 147–156 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bourgeois, N., Escoffier, B., Paschos, V.T.: Fast algorithms for min independent dominating set. In: Patt-Shamir, B., Ekim, T. (eds.) SIROCCO 2010. LNCS, vol. 6058, pp. 247–261. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Bourgeois, N., Croce, F.D., Escoffier, B., Paschos, V.T.: Fast algorithms for min independent dominating set. Discrete Appl. Math. 161(4–5), 558–572 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cheston, G., Fricke, G., Hedetniemi, S., Jacobs, D.: On the computational complexity of upper fractional domination. Discrete Appl. Math. 27(3), 195–207 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cockayne, E.J., Favaron, O., Payan, C., Thomason, A.G.: Contributions to the theory of domination, independence and irredundance in graphs. Discrete Math. 33(3), 249–258 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Courcelle, B., Makowsky, A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theo. Comp. Syst. 33(2), 125–150 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Halldórsson, M.M.: Approximating the minimum maximal independence number. Inf. Process. Lett. 46(4), 169–172 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hare, E.O., Hedetniemi, S.T., Laskar, R.C., Peters, K., Wimer, T.: Linear-time computability of combinatorial problems on generalized-series-parallel graphs. In: Johnson, D.S., et al. (eds.) Discrete Algorithms and Complexity (AP NY) (1987)Google Scholar
  16. 16.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Monographs and Textbooks in Pure and Applied Mathematics, vol. 208. Marcel Dekker, New York (1998)zbMATHGoogle Scholar
  17. 17.
    Henning, M.A., Slater, P.J.: Inequalities relating domination parameters in cubic graphs. Discrete Math. 158(1–3), 87–98 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Håstad, J.: Clique is hard to approximate within \(n^{1-\epsilon }\). Acta Math. 182, 105–142 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hurink, J., Nieberg, T.: Approximating minimum independent dominating sets in wireless networks. Inf. Process. Lett. 109(2), 155–160 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jacobson, M.S., Peters, K.: Chordal graphs and upper irredundance, upper domination and independence. Discrete Math. 86(1–3), 59–69 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kanté, M., Limouzy, V., Mary, A., Nourine, L.: On the enumeration of minimal dominating sets and related notions. SIAM J. Disc. Math. 28(4), 1916–1929 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kanté, M.M., Limouzy, V., Mary, A., Nourine, L., Uno, T.: Polynomial delay algorithm for listing minimal edge dominating sets in graphs. In: Dehne, F., Sack, J.-R., Stege, U. (eds.) WADS 2015. LNCS, vol. 9214, pp. 446–457. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  23. 23.
    Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discrete Appl. Math. 52, 233–252 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lovász, L.: Three short proofs in graph theory. J. Combin. Theory Ser. B 19, 269–271 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43(3), 425–440 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zehavi, M.: Maximum minimal vertex cover parameterized by vertex cover. In: Italiano, G.F., Pighizzini, G., Sannella, D.T. (eds.) MFCS 2015. LNCS, vol. 9235, pp. 589–600. Springer, Heidelberg (2015)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Cristina Bazgan
    • 1
    Email author
  • Ljiljana Brankovic
    • 2
    • 3
  • Katrin Casel
    • 3
  • Henning Fernau
    • 3
  • Klaus Jansen
    • 4
  • Kim-Manuel Klein
    • 4
  • Michael Lampis
    • 1
  • Mathieu Liedloff
    • 5
  • Jérôme Monnot
    • 1
  • Vangelis Th. Paschos
    • 1
  1. 1.Institut Universitaire de FranceUniversité Paris-Dauphine, PSL Research University, CNRS, UMR 7243ParisFrance
  2. 2.The University of NewcastleCallaghanAustralia
  3. 3.Universität TrierTrierGermany
  4. 4.Universität Kiel, Institut für InformatikKielGermany
  5. 5.Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022OrléansFrance

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