Skip to main content
SpringerLink
Log in

The Minimum Entropy Submodular Set Cover Problem

  • Conference paper
  • First Online:
Language and Automata Theory and Applications (LATA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9618))

Included in the following conference series:

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.

G. Istrate and C. Bonchiş were supported by IDEI Grant PN-II-ID-PCE-2011-3-0981. L.P. Dinu was supported by UEFISCDI, PNII-ID-PCE-2011-3-0959.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The problem is NP-hard, rather than NP-complete, since its specification involves general real numbers that may put it outside class NP.

References

  1. Alon, N., Orlitsky, A.: Source coding and graph entropies. IEEE Trans. Inform. Theory 42(5), 1329–1339 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bilbao, J.M.: Cooperative Games on Combinatorial Structures. Kluwer, Boston (2000)

    Book  MATH  Google Scholar 

  3. Bonchiş, C., Istrate, G.: A parametric worst-case approach to fairness in tu-cooperative games. arXiv.org:1208.0283

  4. Cardinal, J., Fiorini, S., Joraet, G.: Tight results on minimum entropy set cover. Algorithmica 51(1), 49–60 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Driessen, T.: Cooperative Games, Solutions and Applications. Kluwer, Boston (1988)

    Book  MATH  Google Scholar 

  6. Dukhovny, A.: General entropy of general measures. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 10(03), 213–225 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fujishige, S.: Submodular Functions and Optimization. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

  8. Fujita, T.: Approximation algorithms for submodular set cover with applications. IEICE Trans. Inf. Syst. E83–D(3), 480–487 (2000)

    Google Scholar 

  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. Wang, X., Jajamovich, G.: Maximum-parsimony haplotype inference based on sparse representations of genotypes. IEEE Trans. Sign. Proc. 60, 2013–2023 (2012)

    Article  MathSciNet  Google Scholar 

  11. Halperin, E., Karp, R.: The minimum entropy set cover problem. Theoret. Comput. Sci. 348(2–3), 340–350 (2005)

    MathSciNet  Google Scholar 

  12. Iwata, S., Orlin, J.B.: A simple combinatorial algorithm for submodular function minimization. J. Comb. Theory Ser. B 84, 1230–1237 (2009)

    MathSciNet  Google Scholar 

  13. Fiorini, S., Cardinal, J., Joret, G.: Minimum entropy orientations. Oper. Res. Lett. 36(6), 680–683 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fiorini, S., Cardinal, J., Joret, G.: Minimum entropy combinatorial optimization problems. Theory Comput. Syst. 51(1), 4–21 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Stanojević, I., Kovačević, M., Šenk, V.: On the entropy of couplings. Inf. Comput. 242, 369–382 (2015)

    Article  Google Scholar 

  16. Madiman, M., Tetali, P.: Information inequalities for joint distributions, with interpretations and applications. IEEE Trans. Inf. Theory 56, 2699–2713 (2010)

    Article  MathSciNet  Google Scholar 

  17. Oxley, J.G.: Matroid Theory. Oxford University Press, Oxford (2006)

    MATH  Google Scholar 

  18. Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory Ser. B 80, 346–355 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Shapley, L.: Cores of convex games. Int. J. Game Theory 1, 11–26 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, B.: Minimum entropy approach to word segmentation problems. Phys. A Stat. Mech. Appl. 293, 583–591 (2001)

    Article  MATH  Google Scholar 

  21. Wolsey, L.: An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2, 385–393 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, West University of Timişoara, Bd. V. Pârvan 4, Timişoara, Romania

    Gabriel Istrate & Cosmin Bonchiş

  2. e-Austria Research Institute, Bd. V. Pârvan 4, cam. 045 B, Timişoara, Romania

    Gabriel Istrate & Cosmin Bonchiş

  3. Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest, Romania

    Liviu P. Dinu

Authors
  1. Gabriel Istrate
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cosmin Bonchiş
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liviu P. Dinu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriel Istrate .

Editor information

Editors and Affiliations

  1. Rovira i Virgili University, Tarragona, Spain

    Adrian-Horia Dediu

  2. Czech Technical University, Prague, Czech Republic

    Jan Janoušek

  3. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  4. Justus Liebig University Giessen, Gießen, Germany

    Bianca Truthe

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Istrate, G., Bonchiş, C., Dinu, L.P. (2016). The Minimum Entropy Submodular Set Cover Problem. In: Dediu, AH., Janoušek, J., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2016. Lecture Notes in Computer Science(), vol 9618. Springer, Cham. https://doi.org/10.1007/978-3-319-30000-9_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-30000-9_23

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-29999-0

  • Online ISBN: 978-3-319-30000-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation