The Minimum Entropy Submodular Set Cover Problem

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9618)

Abstract

We study Minimum Entropy Submodular Set Cover, a variant of the Submodular Set Cover problem (Wolsey [21], Fujito [8], etc.) that generalizes the Minimum Entropy Set Cover problem (Halperin and Karp [11], Cardinal et al. [4]) We give a general bound on the approximation performance of the greedy algorithm using an approach that can be interpreted in terms of a particular type of biased network flows. As an application we rederive known results for the Minimum Entropy Set Cover and Minimum Entropy Orientation problems, and obtain a nontrivial bound for a new problem called the Minimum Entropy Spanning Tree problem. The problem can be applied to (and is partly motivated by) a worst-case approach to fairness in concave cooperative games.

Keywords

Submodular set cover Minimum entropy Approximation algorithms 

References

  1. 1.
    Alon, N., Orlitsky, A.: Source coding and graph entropies. IEEE Trans. Inform. Theory 42(5), 1329–1339 (1996)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Bilbao, J.M.: Cooperative Games on Combinatorial Structures. Kluwer, Boston (2000)CrossRefMATHGoogle Scholar
  3. 3.
    Bonchiş, C., Istrate, G.: A parametric worst-case approach to fairness in tu-cooperative games. arXiv.org:1208.0283
  4. 4.
    Cardinal, J., Fiorini, S., Joraet, G.: Tight results on minimum entropy set cover. Algorithmica 51(1), 49–60 (2008)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Driessen, T.: Cooperative Games, Solutions and Applications. Kluwer, Boston (1988)CrossRefMATHGoogle Scholar
  6. 6.
    Dukhovny, A.: General entropy of general measures. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 10(03), 213–225 (2002)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Fujishige, S.: Submodular Functions and Optimization. Elsevier, Amsterdam (2005)MATHGoogle Scholar
  8. 8.
    Fujita, T.: Approximation algorithms for submodular set cover with applications. IEICE Trans. Inf. Syst. E83–D(3), 480–487 (2000)Google Scholar
  9. 9.
    Bonchiş, C., Istrate, G., Dinu, L.P.: The minimum entropy submodular set cover problem. Manuscript. http://tcs.ieat.ro/wp-content/uploads/2015/10/lata.pdf
  10. 10.
    Wang, X., Jajamovich, G.: Maximum-parsimony haplotype inference based on sparse representations of genotypes. IEEE Trans. Sign. Proc. 60, 2013–2023 (2012)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Halperin, E., Karp, R.: The minimum entropy set cover problem. Theoret. Comput. Sci. 348(2–3), 340–350 (2005)MathSciNetGoogle Scholar
  12. 12.
    Iwata, S., Orlin, J.B.: A simple combinatorial algorithm for submodular function minimization. J. Comb. Theory Ser. B 84, 1230–1237 (2009)MathSciNetGoogle Scholar
  13. 13.
    Fiorini, S., Cardinal, J., Joret, G.: Minimum entropy orientations. Oper. Res. Lett. 36(6), 680–683 (2008)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Fiorini, S., Cardinal, J., Joret, G.: Minimum entropy combinatorial optimization problems. Theory Comput. Syst. 51(1), 4–21 (2012)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Stanojević, I., Kovačević, M., Šenk, V.: On the entropy of couplings. Inf. Comput. 242, 369–382 (2015)CrossRefGoogle Scholar
  16. 16.
    Madiman, M., Tetali, P.: Information inequalities for joint distributions, with interpretations and applications. IEEE Trans. Inf. Theory 56, 2699–2713 (2010)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Oxley, J.G.: Matroid Theory. Oxford University Press, Oxford (2006)MATHGoogle Scholar
  18. 18.
    Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory Ser. B 80, 346–355 (2000)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Shapley, L.: Cores of convex games. Int. J. Game Theory 1, 11–26 (1971)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Wang, B.: Minimum entropy approach to word segmentation problems. Phys. A Stat. Mech. Appl. 293, 583–591 (2001)CrossRefMATHGoogle Scholar
  21. 21.
    Wolsey, L.: An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2, 385–393 (1982)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Mirrokni, V., Abbassi, Z., Thakur, M.: Diversity maximization under matroid constraints. In: Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2013, pp. 32–40. ACM (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Gabriel Istrate
    • 1
    • 2
  • Cosmin Bonchiş
    • 1
    • 2
  • Liviu P. Dinu
    • 3
  1. 1.Department of Computer ScienceWest University of TimişoaraTimişoaraRomania
  2. 2.e-Austria Research InstituteTimişoaraRomania
  3. 3.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharestRomania

Personalised recommendations