Advertisement

Theory of Computing Systems

, Volume 51, Issue 1, pp 4–21 | Cite as

Minimum Entropy Combinatorial Optimization Problems

  • Jean Cardinal
  • Samuel Fiorini
  • Gwenaël Joret
Article

Abstract

We survey recent results on combinatorial optimization problems in which the objective function is the entropy of a discrete distribution. These include the minimum entropy set cover, minimum entropy orientation, and minimum entropy coloring problems.

Keywords

Combinatorial optimization Approximation algorithms NP-hardness Entropy Set cover Graph coloring 

Notes

Acknowledgements

We thank the two anonymous referees for their helpful comments on a previous version of the manuscript. We are especially grateful to one referee for pointing out an error in a previous version of the analysis of the greedy algorithm by dual fitting. This work was supported by the Communauté Française de Belgique (projet ARC).

References

  1. 1.
    Agarwal, S., Belongie, S.: On the non-optimality of four color coding of image partitions. In: ICIP’02: Proceedings of the IEEE International Conference on Image Processing (2002) Google Scholar
  2. 2.
    Alon, N., Orlitsky, A.: Source coding and graph entropies. IEEE Trans. Inf. Theory 42(5), 1329–1339 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bonizzoni, P., Della Vedova, G., Dondi, R., Mariani, L.: Experimental analysis of a new algorithm for partial haplotype completion. Int. J. Bioinform. Res. Appl. 1(4), 461–473 (2005) CrossRefGoogle Scholar
  4. 4.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999) zbMATHCrossRefGoogle Scholar
  5. 5.
    Cardinal, J., Fiorini, S., Joret, G.: Minimum entropy coloring. J. Comb. Optim. 16(4), 361–377 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cardinal, J., Fiorini, S., Joret, G.: Minimum entropy orientations. Oper. Res. Lett. 36, 680–683 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cardinal, J., Fiorini, S., Joret, G.: Tight results on minimum entropy set cover. Algorithmica 51(1), 49–60 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cardinal, J., Fiorini, S., Joret, G.: Minimum entropy combinatorial optimization problems. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds.) Proceedings of the 5th Conference on Computability in Europe, CiE 2009, Heidelberg, Germany, 19–24 July 2009. Lecture Notes in Computer Science, vol. 5635, pp. 79–88. Springer, Berlin (2009) Google Scholar
  9. 9.
    Cardinal, J., Fiorini, S., Joret, G., Jungers, R.M., Munro, J.I.: An efficient algorithm for partial order production. SIAM J. Comput. 39(7), 2927–2940 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Csiszár, I., Körner, J., Lovász, L., Marton, K., Simonyi, G.: Entropy splitting for antiblocking corners and perfect graphs. Combinatorica 10(1), 27–40 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Doshi, V., Shah, D., Médard, M., Jaggi, S.: Graph coloring and conditional graph entropy. In: ACSSC ’06: Proceedings of the Fortieth Asilomar Conference on Signals, Systems and Computers, pp. 2137–2141 (2006) CrossRefGoogle Scholar
  12. 12.
    Doshi, V., Shah, D., Médard, M., Jaggi, S.: Distributed functional compression through graph coloring. In: DCC ’07: Proceedings of the IEEE Data Compression Conference, pp. 93–102 (2007) Google Scholar
  13. 13.
    Doshi, V., Shah, D., Médard, M.: Source coding with distortion through graph coloring. In: ISIT ’07: Proceedings of the IEEE International Symposium on Information Theory, pp. 1501–1505 (2007) Google Scholar
  14. 14.
    Feige, U.: A threshold of lnn for approximating set cover. J. ACM 45(4), 634–652 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Feige, U., Lovász, L., Tetali, P.: Approximating min sum set cover. Algorithmica 40(4), 219–234 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Fukunaga, T., Halldórsson, M.M., Nagamochi, H.: “Rent-or-buy” scheduling and cost coloring problems. In: FSTTCS ’07: Proceedings of the 27th International Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 84–95 (2007) CrossRefGoogle Scholar
  17. 17.
    Fukunaga, T., Halldórsson, M.M., Nagamochi, H.: Robust cost colorings. In: SODA ’08: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1204–1212 (2008) Google Scholar
  18. 18.
    Garey, M.R., Johnson, D.S., Miller, G.L., Papadimitriou, C.H.: The complexity of coloring circular arcs and chords. SIAM J. Algebr. Discrete Methods 1(2), 216–227 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Gijswijt, D., Jost, V., Queyranne, M.: Clique partitioning of interval graphs with submodular costs on the cliques. RAIRO. Rech. Opér. 41(3), 275–287 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gusev, A., Măndoiu, I.I., Paşaniuc, B.: Highly scalable genotype phasing by entropy minimization. IEEE/ACM Trans. Comput. Biol. Bioinform. 5(2), 252–261 (2008) CrossRefGoogle Scholar
  21. 21.
    Halldórsson, M.M., Radhakrishnan, J.: Greed is good: approximating independent sets in sparse and bounded-degree graphs. Algorithmica 18(1), 145–163 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Halperin, E., Karp, R.M.: The minimum-entropy set cover problem. Theor. Comput. Sci. 348(2–3), 240–250 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). Reprint of the 1952 edition zbMATHGoogle Scholar
  24. 24.
    Håstad, J.: Clique is hard to approximate within n 1−ϵ. Acta Math. 182, 105–142 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Kierstead, H.A., Trotter, W.A.: An extremal problem in recursive combinatorics. Congr. Numer. 33, 143–153 (1981) MathSciNetGoogle Scholar
  26. 26.
    Körner, J.: Coding of an information source having ambiguous alphabet and the entropy of graphs. In: Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Tech Univ., Prague, 1971, pp. 411–425. Academia, Prague (1973). Dedicated to the memory of Antonín Špaček Google Scholar
  27. 27.
    Körner, J., Orlitsky, A.: Zero-error information theory. IEEE Trans. Inf. Theory 44(6), 2207–2229 (1998) zbMATHCrossRefGoogle Scholar
  28. 28.
    Koulgi, P., Tuncel, E., Regunathan, S., Rose, K.: On zero-error source coding with decoder side information. IEEE Trans. Inf. Theory 49(1), 99–111 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Marx, D.: A short proof of the NP-completeness of minimum sum interval coloring. Oper. Res. Lett. 33(4), 382–384 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Nicoloso, S., Sarrafzadeh, M., Song, X.: On the sum coloring problem on interval graphs. Algorithmica 23(2), 109–126 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Parnas, M., Ron, D.: Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms. Theor. Comput. Sci. 381(1–3), 183–196 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Pemmaraju, S.V., Raman, R., Varadarajan, K.: Buffer minimization using max-coloring. In: SODA ’04: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 562–571 (2004) Google Scholar
  34. 34.
    Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948) MathSciNetzbMATHGoogle Scholar
  35. 35.
    Simonyi, G.: Graph entropy: a survey. In: Combinatorial Optimization. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 20, pp. 399–441 (1995) Google Scholar
  36. 36.
    Tucker, A.: An efficient test for circular-arc graphs. SIAM J. Comput. 9(1), 1–24 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Yannakakis, M., Gavril, F.: The maximum k-colorable subgraph problem for chordal graphs. Inf. Process. Lett. 24(2), 133–137 (1987) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  2. 2.Département de MathématiqueUniversité Libre de BruxellesBrusselsBelgium

Personalised recommendations