Skip to main content
SpringerLink
Log in
Book cover

International Conference on Combinatorial Optimization and Applications

COCOA 2011: Combinatorial Optimization and Applications pp 1–15Cite as

The Complexity of Testing Monomials in Multivariate Polynomials

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6831))

Abstract

The work in this paper is to initiate a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by certain economically compact structure has a multilinear monomial in its sum-product expansion. The complexity aspects of this problem and its variants are investigated with two objectives. One is to understand how this problem relates to critical problems in complexity, and if so to what extent. The other is to exploit possibilities of applying algebraic properties of polynomials to the study of those problems. A series of results about ΠΣΠ and ΠΣ polynomials are obtained in this paper, laying a basis for further study along this line.

This is a preview of subscription content, 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 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Manindra, A., Neeraj, K., Nitin, S.: PRIMES is in P. Ann. of Math. 160(2), 781–793 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the ACM 45(3), 501–555 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aspvall, B., Plass, M.F., Tarjan, R.E.: A linear-time algorithm for testing the truth of certain quantified boolean formulas. Information Processing Letters 8(3), 121–123 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beigel, R.: The polynomial method in circuit complexity. In: Proceedings of the Eighth Conference on Structure in Complexity Theory, pp. 82–95 (1993)

    Google Scholar 

  5. Bshouty, N.H., Chen, Z., Decatur, S.E., Homer, S.: One the learnability of Z N -DNF formulas. In: Proceedings of the Eighth Annual Conference on Computational Learning Theory (COLT 1995), pp. 198–205. ACM, Santa Cruz (1995)

    Chapter  Google Scholar 

  6. Chen, Z., Fu, B.: Approximating multilinear monomial coefficients and maximum multilinear monomials in multivariate polynomials. In: Wu, W., Daescu, O. (eds.) COCOA 2010, Part I. LNCS, vol. 6508, pp. 309–323. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  7. Chen, Z., Fu, B., Liu, Y., Schweller, R.T.: Algorithms for Testing Monomials in Multivariate Polynomials. In: Proceedings of the Fifth International Conference on Combinatorial Optimization and Applications (2011)

    Google Scholar 

  8. Feige, U., Goldwasser, S., Lovász, L., Safra, S., Szegedy, M.: Interactive proofs and the hardness of approximating cliques. Journal of the ACM 43(2), 268–292 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fu, B.: Separating PH from PP by relativization. Acta Math. Sinica 8(3), 329–336 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hopcroft, J.E., Karp, R.M.: An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing 2(4), 225–231 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kabanets, V., Impagliazzo, R.: Derandomizing polynomial identity tests means proving circuit lower bounds. In: STOC, pp. 355–364 (2003)

    Google Scholar 

  12. Klivans, A., Servedio, R.A.: Learning DNF in time \(2^{\tilde{O}(n^{1/3})}\). In: STOC, pp. 258–265 (2001)

    Google Scholar 

  13. Koutis, I.: Faster algebraic algorithms for path and packing problems. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 575–586. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  14. Minsky, M., Papert, S.: Perceptrons (expanded edition 1988). MIT Press, Cambridge (1968)

    Google Scholar 

  15. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  16. Robson, J.M.: Algorithms for maximum independent sets. Journal of Algorithms 7(3), 425–440 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shamir, A.: IP = PSPACE. Journal of the ACM 39(4), 869–877 (1992)

    Article  MathSciNet  Google Scholar 

  18. Williams, R.: Finding paths of length k in O *(2k) time. Information Processing Letters 109, 315–318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Texas-Pan American, Edinburg, TX, 78539, USA

    Zhixiang Chen & Bin Fu

Authors
  1. Zhixiang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bin Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Zhejiang Normal University, 688 Yingbin Road, 321004, Jinhua, Zhejiang Province, China

    Weifan Wang  & Xuding Zhu  & 

  2. Department of Computer Science, University of Texas at Dallas, 75080, Richardson, TX, USA

    Ding-Zhu Du

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Chen, Z., Fu, B. (2011). The Complexity of Testing Monomials in Multivariate Polynomials. In: Wang, W., Zhu, X., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2011. Lecture Notes in Computer Science, vol 6831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22616-8_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-22616-8_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22615-1

  • Online ISBN: 978-3-642-22616-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation