Advertisement

Maximizing a Submodular Function with Viability Constraints

  • Wolfgang Dvořák
  • Monika Henzinger
  • David P. Williamson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8125)

Abstract

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 NP-hard, we investigate approximation algorithm. 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 + ε)-approximation algorithm for our problem setting (even for additive functions) and that there is no approximation algorithm for a slight extension of this setting.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bordewich, M., Semple, C.: Nature reserve selection problem: A tight approximation algorithm. IEEE/ACM Transactions on Computational Biology and Bioinformatics 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. Journal of Mathematical Biology 64(1-2), 69–85 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Faith, D.P.: Faith. Conservation evaluation and phylogenetic diversity. Biological Conservation 61(1), 1–10 (1992)CrossRefGoogle Scholar
  4. 4.
    Faller, B., Semple, C., Welsh, D.: Optimizing Phylogenetic Diversity with Ecological Constraints. Annals of Combinatorics 15, 255–266 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions – II. Mathematical Programming Study 8, 73–87 (1978)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Goundan, P.R., Schulz, A.S.: Revisiting the greedy approach to submodular set function maximization. Working Paper, Massachusetts Institute of Technology (2007), http://www.optimization-online.org/DB_HTML/2007/08/1740.html
  8. 8.
    Khuller, S., Moss, A., Naor, J.: The budgeted maximum coverage problem. Inf. Process. Lett. 70(1), 39–45 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    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, pp. 323–332. ACM (2009)Google Scholar
  10. 10.
    Moulton, V., Semple, C., Steel, M.: Optimizing phylogenetic diversity under constraints. Journal of Theoretical Biology 246(1), 186–194 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions — I. Mathematical Programming 14, 265–294 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Pardi, F., Goldman, N.: Species choice for comparative genomics: being greedy works. PLoS Genetics 71, 71 (2005)Google Scholar
  13. 13.
    Hsu, T.S., Tsai, K.-H., Wang, D.-W., Lee, D.T.: Two variations of the minimum steiner problem. J. Comb. Optim. 9(1), 101–120 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Steel, M.: Phylogenetic diversity and the greedy algorithm. Systematic Biology 54(4), 527–529 (2005)CrossRefGoogle Scholar
  15. 15.
    Vondrák, J.: Submodular functions and their applications. In: SODA 2013 Plenary Talk (2013) Slides available at, http://theory.stanford.edu/~jvondrak/data/SODA-plenary-talk.pdf
  16. 16.
    Weitzman, M.L.: The Noah’s ark problem. Econometricay 66, 1279–1298 (1998)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Wolfgang Dvořák
    • 1
  • Monika Henzinger
    • 1
  • David P. Williamson
    • 2
  1. 1.Fakultät für InformatikUniversität WienViennaAustria
  2. 2.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA

Personalised recommendations