, Volume 77, Issue 1, pp 152–172 | Cite as

Maximizing a Submodular Function with Viability Constraints

  • Wolfgang Dvořák
  • Monika Henzinger
  • David P. Williamson


We study the problem of maximizing a monotone submodular function with viability constraints. This problem originates from computational biology, where we are given a phylogenetic tree over a set of species and a directed graph, the so-called food web, encoding viability constraints between these species. These food webs usually have constant depth. The goal is to select a subset of k species that satisfies the viability constraints and has maximal phylogenetic diversity. As this problem is known to be \(\mathsf{NP}\)-hard, we investigate approximation algorithms. We present the first constant factor approximation algorithm if the depth is constant. Its approximation ratio is \((1-\frac{1}{\sqrt{e}})\). This algorithm not only applies to phylogenetic trees with viability constraints but for arbitrary monotone submodular set functions with viability constraints. Second, we show that there is no \((1-1/e+\epsilon )\)-approximation algorithm for our problem setting (even for additive functions) and that there is no approximation algorithm for a slight extension of this setting.


Approximation algorithms Submodular functions Phylogenetic diversity Viability constraints 



The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 340506.


  1. 1.
    Bordewich, M., Semple, C.: Nature reserve selection problem: a tight approximation algorithm. IEEE/ACM Trans. Comput. Biol. Bioinform. 5(2), 275–280 (2008)CrossRefGoogle Scholar
  2. 2.
    Bordewich, M., Semple, C.: Budgeted nature reserve selection with diversity feature loss and arbitrary split systems. J. Math. Biol. 64(1–2), 69–85 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chernomor, O., Minh, B.Q., Forest, F., Klaere, S., Ingram, T., Henzinger, M., von Haeseler, A.: Split diversity in constrained conservation prioritization using integer linear programming. Methods Ecol. Evol. 6(1), 83–91 (2015)CrossRefGoogle Scholar
  4. 4.
    Dvořák, W., Henzinger, M., Williamson, D.P.: Maximizing a Submodular Function with Viability Constraints. In: Bodlaender H.L., Italiano G.F. (eds.), Algorithms—ESA 2013—Proceedings of the 21st Annual European Symposium, Sophia Antipolis, France, September 2–4, 2013. Lecture Notes in Computer Science, vol. 8125, pp. 409–420. Springer (2013)Google Scholar
  5. 5.
    Faith, D.P.: Conservation evaluation and phylogenetic diversity. Biol. Conserv. 61(1), 1–10 (1992)CrossRefGoogle Scholar
  6. 6.
    Faller, B., Semple, C., Welsh, D.: Optimizing phylogenetic diversity with ecological constraints. Ann. Comb. 15, 255–266 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions—II. Math. Program. Study 8, 73–87 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Goundan, P.R., Schulz, A.S.: Revisiting the Greedy Approach to Submodular Set Function Maximization. In: Working Paper, Massachusetts Institute of Technology, 2007.
  10. 10.
    Hsu, T., Tsai, K.H., Wang, D.W., Lee, D.T.: Two variations of the minimum Steiner problem. J. Comb. Optim. 9(1), 101–120 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Khuller, S., Moss, A., Naor, J.: The budgeted maximum coverage problem. Inf. Process. Lett. 70(1), 39–45 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lee, J., Mirrokni, V.S., Nagarajan, V., Sviridenko, M.: Non-monotone Submodular Maximization Under Matroid and Knapsack Constraints. In: Mitzenmacher M. (ed.) Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31–June 2 (2009)Google Scholar
  13. 13.
    Moulton, V., Semple, C., Steel, M.: Optimizing phylogenetic diversity under constraints. J. Theor. Biol. 246(1), 186–194 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions—I. Math. Program. 14, 265–294 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pardi, F., Goldman, N.: Species choice for comparative genomics: being greedy works. PLoS Genet. 1(6), e71 (2005)CrossRefGoogle Scholar
  16. 16.
    Petrank, E.: The hardness of approximation: gap location. Comput. Complex. 4, 133–157 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Steel, M.: Phylogenetic diversity and the greedy algorithm. Syst. Biol. 54(4), 527–529 (2005)CrossRefGoogle Scholar
  18. 18.
    van der Heide, C.M., van den Bergh, J., van Ierland, E.: Extending weitzman’s economic ranking of biodiversity protection: combining ecological and genetic considerations. Ecol. Econ. 55(2), 218–223 (2005)CrossRefGoogle Scholar
  19. 19.
    Vondrák, J.: Submodular Functions and their Applications. In: SODA 2013 Plenary Talk.
  20. 20.
    Weitzman, M.L.: The Noah’s ark problem. Econometricay 66, 1279–1298 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Fakultät für InformatikUniversität WienViennaAustria
  2. 2.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA

Personalised recommendations