Advertisement

Some Complete and Intermediate Polynomials in Algebraic Complexity Theory

  • Meena Mahajan
  • Nitin Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)

Abstract

We provide a list of new natural \(\mathsf {VNP}\)-Intermediate polynomial families, based on basic (combinatorial) \(\mathsf {NP}\)-Complete problems that are complete under parsimonious reductions. Over finite fields, these families are in \(\mathsf {VNP}\), and under the plausible hypothesis \(\mathsf {Mod}_p\mathsf {P}\not \subseteq \mathsf {P/poly}\), are neither \(\mathsf {VNP}\)-hard (even under oracle-circuit reductions) nor in \(\mathsf {VP}\). Prior to this, only the Cut Enumerator polynomial was known to be \(\mathsf {VNP}\)-intermediate, as shown by Bürgisser in 2000.

We next show that over rationals and reals, two of our intermediate polynomials, based on satisfiability and Hamiltonian cycle, are not monotone affine polynomial-size projections of the permanent. This augments recent results along this line due to Grochow.

Finally, we describe a (somewhat natural) polynomial defined independent of a computation model, and show that it is \(\mathsf {VP}\)-complete under polynomial-size projections. This complements a recent result of Durand et al. (2014) which established \(\mathsf {VP}\)-completeness of a related polynomial but under constant-depth oracle circuit reductions. Both polynomials are based on graph homomorphisms. A simple restriction yields a family similarly complete for \(\mathsf {VBP}\).

Keywords

Hamiltonian Cycle Tree Decomposition Arithmetic Circuit Graph Homomorphism Affine Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Alon, N., Boppana, R.B.: The monotone circuit complexity of Boolean functions. Combinatorica 7(1), 1–22 (1987)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Avis, D., Tiwary, H.R.: On the extension complexity of combinatorial polytopes. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 57–68. Springer, Heidelberg (2013)Google Scholar
  3. 3.
    Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory. Algorithms and Computation in Mathematics, vol. 7. Springer, Heidelberg (2000)MATHGoogle Scholar
  4. 4.
    Bürgisser, P.: On the structure of Valiant’s complexity classes. Discrete Math. Theor. Comput. Sci. 3(3), 73–94 (1999)MathSciNetMATHGoogle Scholar
  5. 5.
    Bürgisser, P.: Cook’s versus Valiant’s hypothesis. Theor. Comput. Sci. 235(1), 71–88 (2000)CrossRefMATHGoogle Scholar
  6. 6.
    Capelli, F., Durand, A., Mengel, S.: The arithmetic complexity of tensor contractions. In: Symposium on Theoretical Aspects of Computer Science STACS. LIPIcs, vol. 20, pp. 365–376 (2013)Google Scholar
  7. 7.
    Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Switzerland (2015)CrossRefMATHGoogle Scholar
  8. 8.
    de Rugy-Altherre, N.: A dichotomy theorem for homomorphism polynomials. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 308–322. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Díaz, J., Serna, M.J., Thilikos, D.M.: Counting h-colorings of partial k-trees. Theor. Comput. Sci. 281(1–2), 291–309 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Durand, A., Mahajan, M., Malod, G., de Rugy-Altherre, N., Saurabh, N.: Homomorphism polynomials complete for VP. In: 34th Foundation of Software Technology and Theoretical Computer Science Conference, FSTTCS, pp. 493–504 (2014)Google Scholar
  11. 11.
    Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17(3), 449–467 (1965)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., Wolf, R.D.: Exponential lower bounds for polytopes in combinatorial optimization. J. ACM 62(2), 17 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)MATHGoogle Scholar
  14. 14.
    Grochow, J.A.: Monotone projection lower bounds from extended formulation lower bounds. [cs.CC] (2015). arXiv:1510.08417
  15. 15.
    Hell, P., Nešetřil, J.: Graphs And Homomorphisms. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2004)CrossRefMATHGoogle Scholar
  16. 16.
    Jerrum, M., Snir, M.: Some exact complexity results for straight-line computations over semirings. J. ACM 29(3), 874–897 (1982)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jukna, S.: Why is Hamilton cycle so different from permanent? (2014). http://cstheory.stackexchange.com/questions/27496/why-is-hamiltonian-cycle-so-different-from-permanent
  18. 18.
    Karp, R.M., Lipton, R.: Turing machines that take advice. L’enseignement mathématique 28(2), 191–209 (1982)MathSciNetMATHGoogle Scholar
  19. 19.
    Ladner, R.E.: On the structure of polynomial time reducibility. J. ACM 22(1), 155–171 (1975)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Malod, G., Portier, N.: Characterizing Valiant’s algebraic complexity classes. J. Complex. 24(1), 16–38 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Mengel, S.: Characterizing arithmetic circuit classes by constraint satisfaction problems. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 700–711. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  22. 22.
    Raz, R.: Elusive functions and lower bounds for arithmetic circuits. Theor. Comput. 6, 135–177 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Razborov, A.A.: Lower bounds on the monotone complexity of some Boolean functions. Dokl. Akad. Nauk SSSR 281(4), 798–801 (1985)MathSciNetMATHGoogle Scholar
  24. 24.
    Razborov, A.A.: Lower bounds on monotone complexity of the logical permanent. Math. Notes Acad. Sci. USSR 37(6), 485–493 (1985)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Rothvoß, T.: The matching polytope has exponential extension complexity. In: Symposium on Theory of Computing, STOC, pp. 263–272. New York, 31 May–03 June 2014Google Scholar
  26. 26.
    Simon, J.: On the difference between one and many. In: Salomaa, A., Steinby, M. (eds.) Automata, Languages and Programming. LNCS, vol. 52, pp. 480–491. Springer, Heidelberg (1977)CrossRefGoogle Scholar
  27. 27.
    Valiant, L.G.: Completeness classes in algebra. In: Symposium on Theory of Computing STOC, pp. 249–261 (1979)Google Scholar
  28. 28.
    Valiant, L.G., Skyum, S., Berkowitz, S., Rackoff, C.: Fast parallel computation of polynomials using few processors. SIAM J. Comput. 12(4), 641–644 (1983)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.The Institute of Mathematical SciencesChennaiIndia

Personalised recommendations