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Hardness results for approximate pure Horn CNF formulae minimization

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Abstract

We study the hardness of approximation of clause minimum and literal minimum representations of pure Horn functions in n Boolean variables. We show that unless P=NP, it is not possible to approximate in polynomial time the minimum number of clauses and the minimum number of literals of pure Horn CNF representations to within a factor of \(2^{\log^{1-o(1)} n}\). This is the case even when the inputs are restricted to pure Horn 3-CNFs with O(n 1+ε) clauses, for some small positive constant ε. Furthermore, we show that even allowing sub-exponential time computation, it is still not possible to obtain constant factor approximations for such problems unless the Exponential Time Hypothesis turns out to be false.

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References

  1. Arora, S., Barak, B.: Computational Complexity: A modern approach. Cambridge University Press. xxiv+579. ISBN: 978-0-521-42426-4 (2009)

  2. Arora, S., Babai, L., Stern, J., Sweedyk, Z.: The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. Syst. Sci. 54(2, part 2), 317–331 (1997). doi:10.1006/jcss.1997.1472

    Article  MathSciNet  MATH  Google Scholar 

  3. Arora, S., Lund, C.: Hardness of approximations. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-Hard Problems. PWS Publishing, Boston (1996)

  4. Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. J. ACM 45(3), 501–555 (1998). doi:10.1145/278298.278306

    Article  MathSciNet  MATH  Google Scholar 

  5. Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. J. ACM 45(1), 70–122 (1998). doi:10.1145/273865.273901

    Article  MathSciNet  MATH  Google Scholar 

  6. Ausiello, G., D’Atri, A., Saccà, D.: Minimal representation of directed hypergraphs. SIAM J. Comput. 15(2), 418–431 (1986). doi:10.1137/0215029

    Article  MathSciNet  MATH  Google Scholar 

  7. Bhattacharya, A., DasGupta, B., Mubayi, D., Turán, G.: On approximate Horn formula minimization. In: Abramsky, S., Gavoille, C., Kirchner, C., auf der Heide, F.M., Spirakis, P.G. (eds.) ICALP (1), Lecture Notes in Computer Science, vol. 6198, pp. 438–450. Springer (2010)

  8. Boros, E., Ċepek, O., Kuċera, P.: A decomposition method for CNF minimality proofs. Theor. Comput. Sci. 510, 111–126 (2013). doi:10.1016/j.tcs.2013.09.016

    Article  MATH  Google Scholar 

  9. Boros, E., Ċepek, O., Kogan, A., Kuċera, P.: Exclusive and essential sets of implicates of Boolean functions. Discret. Appl. Math. 158(2), 81–96 (2010). doi:10.1016/j.dam.2009.08.012

    Article  MATH  Google Scholar 

  10. Boros, E., Gruber, A.: Hardness results for approximate pure Horn CNF formulae minimization. In: International Symposium on Artificial Intelligence and Mathematics (ISAIM 2012), Fort Lauderdale, Florida, USA, January 9-11 (2012)

  11. Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the 3rd Annual ACM Symposium on Theory of Computing (STOC). pp. 151-158. Ohio. doi:10.1145/800157.805047 (1971)

  12. Crama, Y., Hammer, P.L. (eds.): Boolean Functions: Theory, Algorithms, and Applications. Encyclopedia of Mathematics and its Applications, vol. 142. Cambridge University Press, Cambridge (2011)

  13. Cormen, T. H., Leiserson, C. E., Rivest, R.L., Stein, C.: Introduction to algorithms, 3rd edn, pp. xx+1292. MIT Press, Cambridge. ISBN 978-0-262-03384-8 (2009)

  14. Dinur, I., Harsha, P.: Composition of low-error 2-query PCPs using decodable PCPs. SIAM J. Comput. 42(6), 2452–2486 (2013). doi:10.1137/100788161

    Article  MathSciNet  MATH  Google Scholar 

  15. Dinur, I., Safra, S.: On the hardness of approximating label-cover. Inf. Process. Lett. 89(5), 247–254 (2004). doi:10.1016/j.ipl.2003.11.007

    Article  MathSciNet  MATH  Google Scholar 

  16. Dowling, W.F., Gallier, J.H.: Linear-time algorithms for testing the satisfiability of propositional Horn formulae. J. Log. Prog. 1(3), 267–284 (1984). doi:10.1016/0743-1066(84)90014-1

    Article  MathSciNet  MATH  Google Scholar 

  17. Feige, U., Goldwasser, S., Lovász, L., Safra, S., Szegedy, M.: Interactive proofs and the hardness of approximating cliques. J. ACM 43(2), 268–292 (1996). doi:10.1145/226643.226652

    Article  MathSciNet  MATH  Google Scholar 

  18. Feige, U., Lovász, L.: Two-prover one-round proof systems: Their power and their problems (extended abstract). STOC, pp. 733–744. ACM (1992)

  19. Garey, M.R., Johnson, D.S. Computers and intractability. W. H. Freeman and Co., San Francisco. A guide to the theory of NP-completeness, A Series of Books in the Mathematical Sciences (1979)

  20. Hammer, P.L., Kogan, A.: Horn functions and their DNFs. Inf. Process. Lett. 44(1), 23–29 (1992). doi:10.1016/0020-0190(92)90250-Y

    Article  MathSciNet  MATH  Google Scholar 

  21. Hammer, P.L., Kogan, A.: Optimal compression of propositional Horn knowledge bases: complexity and approximation. Artif. Intell. 64(1), 131–145 (1993). doi:10.1016/0004-3702(93)90062-G

    Article  MathSciNet  MATH  Google Scholar 

  22. Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001). doi:10.1006/jcss.2000.1727. Special issue on the fourteenth annual IEEE conference on computational complexity (Atlanta, GA, 1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Itai, A., Makowsky, J.A.: Unification as a complexity measure for logic programming. J. Log. Prog. 4(2), 105–117 (1987). doi:10.1016/0743-1066(87)90014-8

    Article  MathSciNet  MATH  Google Scholar 

  24. Kortsarz, G.: On the hardness of approximating spanners. Algorithmica 30(3), 432–450 (2001). Approximation algorithms for combinatorial optimization problems

    Article  MathSciNet  MATH  Google Scholar 

  25. Maier, D.: Minimum covers in relational database model. J. ACM 27(4), 664–674 (1980). doi:10.1145/322217.322223

    Article  MathSciNet  MATH  Google Scholar 

  26. Minoux, M.: Ltur: A simplified linear-time unit resolution algorithm for Horn formulae and computer implementation. Inf. Process. Lett. 29(1), 1–12 (1988). doi:10.1016/0020-0190(88)90124-X

    Article  MathSciNet  MATH  Google Scholar 

  27. Moshkovitz, D., Raz, R.: Two-query PCP with subconstant error. J. ACM 57(5), Art, 29, 29 (2010). doi:10.1145/1.754399.1754402

    Article  MathSciNet  Google Scholar 

  28. Umans, C.: Hardness of approximating \({\Sigma ^{p}_{2}}\) minimization problems. In: 40th Annual Symposium on Foundations of Computer Science (New York, 1999), pp. 465–474, IEEE Computer Soc., Los Alamitos. doi:10.1109/SFFCS.1999.814619 (1999)

  29. Vazirani, V.V.: Approximation Algorithms. Springer (2001)

  30. Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. RUTCOR and MSIS, Rutgers University, 100 Rockafeller Road, Piscataway, NJ, 08854-8003, USA

    Endre Boros & Aritanan Gruber

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  1. Endre Boros
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  2. Aritanan Gruber
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Correspondence to Aritanan Gruber.

Additional information

The authors gratefully acknowledge the partial support by NSF grants IIS 0803444 and by CMMI 0856663. The second author also gratefully acknowledge the partial support by the joint CAPES (Brazil)/Fulbright (USA) fellowship process BEX-2387050/15061676.

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Boros, E., Gruber, A. Hardness results for approximate pure Horn CNF formulae minimization. Ann Math Artif Intell 71, 327–363 (2014). https://doi.org/10.1007/s10472-014-9415-9

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