The Minimum-Entropy Set Cover Problem

  • Eran Halperin
  • Richard M. Karp
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3142)


We consider the minimum entropy principle for learning data generated by a random source and observed with random noise.

In our setting we have a sequence of observations of objects drawn uniformly at random from a population. Each object in the population belongs to one class. We perform an observation for each object which determines that it belongs to one of a given set of classes. Given these observations, we are interested in assigning the most likely class to each of the objects.

This scenario is a very natural one that appears in many real life situations. We show that under reasonable assumptions finding the most likely assignment is equivalent to the following variant of the set cover problem. Given a universe U and a collection \({\cal S} = (S_1,\ldots,S_m)\) of subsets of U, we wish to find an assignment \(f:U \to \cal S\) such that uf(u) and the entropy of the distribution defined by the values |f− − 1(Si)| is minimized.

We show that this problem is NP-hard and that the greedy algorithm for set cover finds a cover with an additive constant error with respect to the optimal cover. This sheds a new light on the behavior of the greedy set cover algorithm. We further enhance the greedy algorithm and show that the problem admits a polynomial time approximation scheme (PTAS).

Finally, we demonstrate how this model and the greedy algorithm can be useful in real life scenarios, and in particular, in problems arising naturally in computational biology.


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  1. 1.
    Chvátal, V.: A greedy heuristic for the set-covering problem. Mathematics of Operations Research 4, 233–235 (1979)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45 (1998)Google Scholar
  3. 3.
    Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1934)Google Scholar
  4. 4.
    Herskovits, E.H., Cooper, G.F.: Kutato: an entropy-driven system for construction of probabilistic expert systems from database. In: Proceedings of the Sixth Conference on Uncertainty in Artificial Intelligence, pp. 54–62 (1990)Google Scholar
  5. 5.
    Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. In: Proceedings of the 25rd Annual ACM Symposium on Theory of Computing, San Diego, California, pp. 286–293 (1993)Google Scholar
  6. 6.
    Ran Raz and Shmuel Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, El Paso, Texas, pp. 475–484 (1997)Google Scholar
  7. 7.
    Roberts, S., Everson, R., Rezek, I.: Minimum entropy data partitioning. In: Proc. of 9th International Conference on Articial Neural Networks, pp. 844–849 (1999)Google Scholar
  8. 8.
    Roberts, S.J., Holmes, C., Denison, D.: Minimum-entropy data partitioning using reversible jump markov chain monte carlo. IEEE Transactions on Pattern Analysis and Machine Intelligence 23(8), 909–914 (2001)CrossRefGoogle Scholar
  9. 9.
    Sharan, R.: Personal communication (2003)Google Scholar
  10. 10.
    Xiang, Y., Michael Wong, S.K., Cercone, N.: A “microscopic” study of minimum entropy search in learning decomposable markov networks. Machine Learning 26(1), 65–92 (1997)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Eran Halperin
    • 1
  • Richard M. Karp
    • 2
  1. 1.CS departmentPrinceton UniversityPrincetonUSA
  2. 2.International Computer Science InstituteBerkeleyUSA

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