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Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators

Authors V. Arvind, Abhranil Chatterjee, Rajit Datta, Partha Mukhopadhyay

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V. Arvind
  • Institute of Mathematical Sciences (HBNI), Chennai, India
Abhranil Chatterjee
  • Institute of Mathematical Sciences (HBNI), Chennai, India
Rajit Datta
  • Chennai Mathematical Institute, Chennai, India
Partha Mukhopadhyay
  • Chennai Mathematical Institute, Chennai, India

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V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.FSTTCS.2018.7

Abstract

Let F[X] be the polynomial ring over the variables X={x_1,x_2, ..., x_n}. An ideal I= <p_1(x_1), ..., p_n(x_n)> generated by univariate polynomials {p_i(x_i)}_{i=1}^n is a univariate ideal. We study the ideal membership problem for the univariate ideals and show the following results. - Let f(X) in F[l_1, ..., l_r] be a (low rank) polynomial given by an arithmetic circuit where l_i : 1 <= i <= r are linear forms, and I=<p_1(x_1), ..., p_n(x_n)> be a univariate ideal. Given alpha in F^n, the (unique) remainder f(X) mod I can be evaluated at alpha in deterministic time d^{O(r)} * poly(n), where d=max {deg(f),deg(p_1)...,deg(p_n)}. This yields a randomized n^{O(r)} algorithm for minimum vertex cover in graphs with rank-r adjacency matrices. It also yields an n^{O(r)} algorithm for evaluating the permanent of a n x n matrix of rank r, over any field F. Over Q, an algorithm of similar run time for low rank permanent is due to Barvinok [Barvinok, 1996] via a different technique. - Let f(X)in F[X] be given by an arithmetic circuit of degree k (k treated as fixed parameter) and I=<p_1(x_1), ..., p_n(x_n)>. We show that in the special case when I=<x_1^{e_1}, ..., x_n^{e_n}>, we obtain a randomized O^*(4.08^k) algorithm that uses poly(n,k) space. - Given f(X)in F[X] by an arithmetic circuit and I=<p_1(x_1), ..., p_k(x_k)>, membership testing is W[1]-hard, parameterized by k. The problem is MINI[1]-hard in the special case when I=<x_1^{e_1}, ..., x_k^{e_k}>.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Combinatorial Nullstellensatz
  • Ideal Membership
  • Parametric Hardness
  • Low Rank Permanent

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References

  1. Noga Alon. Combinatorial Nullstellensatz. Comb. Probab. Comput., 8(1-2):7-29, January 1999. URL: http://dl.acm.org/citation.cfm?id=971651.971653.
  2. Kotlov Andrew and Lovász László. The rank and size of graphs. Journal of Graph Theory, 23(2):185-189, 1996. Google Scholar
  3. Vikraman Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Fast Exact Algorithms Using Hadamard Product of Polynomials. CoRR, abs/1807.04496, 2018. URL: http://arxiv.org/abs/1807.04496.
  4. Vikraman Arvind and Partha Mukhopadhyay. The ideal membership problem and polynomial identity testing. Inf. Comput., 208(4):351-363, 2010. URL: http://dx.doi.org/10.1016/j.ic.2009.06.003.
  5. Alexander I. Barvinok. Two Algorithmic Results for the Traveling Salesman Problem. Math. Oper. Res., 21(1):65-84, February 1996. URL: http://dx.doi.org/10.1287/moor.21.1.65.
  6. Andreas Björklund, Petteri Kaski, and Lukasz Kowalik. Constrained Multilinear Detection and Generalized Graph Motifs. Algorithmica, 74(2):947-967, 2016. URL: http://dx.doi.org/10.1007/s00453-015-9981-1.
  7. Cornelius Brand, Holger Dell, and Thore Husfeldt. Extensor-coding. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 151-164, 2018. URL: http://dx.doi.org/10.1145/3188745.3188902.
  8. George E. Collins. Subresultants and Reduced Polynomial Remainder Sequences. J. ACM, 14(1):128-142, 1967. URL: http://dx.doi.org/10.1145/321371.321381.
  9. David A. Cox, John Little, and Donal O'Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics). Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2007. Google Scholar
  10. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
  11. Richard A. Demillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Information Processing Letters, 7(4):193-195, 1978. URL: http://dx.doi.org/10.1016/0020-0190(78)90067-4.
  12. Rodney G. Downey, Vladimir Estivill-Castro, Michael R. Fellows, Elena Prieto-Rodriguez, and Frances A. Rosamond. Cutting Up is Hard to Do: the Parameterized Complexity of k-Cut and Related Problems. Electr. Notes Theor. Comput. Sci., 78:209-222, 2003. URL: http://dx.doi.org/10.1016/S1571-0661(04)81014-4.
  13. Neeraj Kayal and Nitin Saxena. Polynomial Identity Testing for Depth 3 Circuits. Computational Complexity, 16(2):115-138, 2007. URL: http://dx.doi.org/10.1007/s00037-007-0226-9.
  14. Pascal Koiran. Hilbert’s Nullstellensatz Is in the Polynomial Hierarchy. J. Complexity, 12(4):273-286, 1996. URL: http://dx.doi.org/10.1006/jcom.1996.0019.
  15. Andrev#-#-#ı Kotlov. Rank and Chromatic Number of a Graph. J. Graph Theory, 26(1):1-8, September 1997. Google Scholar
  16. Ioannis Koutis. Faster Algebraic Algorithms for Path and Packing Problems. In Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part I: Tack A: Algorithms, Automata, Complexity, and Games, pages 575-586, 2008. URL: http://dx.doi.org/10.1007/978-3-540-70575-8_47.
  17. Ioannis Koutis. Constrained multilinear detection for faster functional motif discovery. Inf. Process. Lett., 112(22):889-892, 2012. URL: http://dx.doi.org/10.1016/j.ipl.2012.08.008.
  18. Ioannis Koutis and Ryan Williams. LIMITS and applications of group algebras for parameterized problems. ACM Trans. Algorithms, 12(3):31:1-31:18, 2016. URL: http://dx.doi.org/10.1145/2885499.
  19. Hwangrae Lee. Power Sum Decompositions of Elementary Symmetric Polynomials. Linear Algebra and its Applications, 492:89 - 97, 2016. URL: http://dx.doi.org/10.1016/j.laa.2015.11.018.
  20. K. Mahler. An inequality for the discriminant of a polynomial. Michigan Math. J., 11(3):257-262, September 1964. URL: http://dx.doi.org/10.1307/mmj/1028999140.
  21. E. Mayr and A. Meyer. The complexity of word problem for commutative semigroups and polynomial ideals. Adv. Math, 46:305-329, 1982. Google Scholar
  22. Kevin Pratt. Faster Algorithms via Waring Decompositions. CoRR, abs/1807.06194, 2018. URL: http://arxiv.org/abs/1807.06194.
  23. Sanjeev Saluja. A note on the permanent value problem. Information Processing Letters, 43(1):1-5, 1992. URL: http://dx.doi.org/10.1016/0020-0190(92)90021-M.
  24. Nitin Saxena and C. Seshadhri. From sylvester-gallai configurations to rank bounds: Improved blackbox identity test for depth-3 circuits. J. ACM, 60(5):33:1-33:33, 2013. URL: http://dx.doi.org/10.1145/2528403.
  25. Thomas J. Schaefer. The Complexity of Satisfiability Problems. In Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, STOC '78, pages 216-226, New York, NY, USA, 1978. ACM. Google Scholar
  26. Jacob T. Schwartz. Fast Probabilistic algorithm for verification of polynomial identities. J. ACM., 27(4):701-717, 1980. Google Scholar
  27. Madhu Sudan. Lectures on Algebra and Computation: Lecture Notes 6,12,13,14, 1998. Google Scholar
  28. Ryan Williams. Finding paths of length k in O^*(2^k) time. Inf. Process. Lett., 109(6):315-318, 2009. URL: http://dx.doi.org/10.1016/j.ipl.2008.11.004.
  29. R. Zippel. Probabilistic algorithms for sparse polynomials. In Proc. of the Int. Sym. on Symbolic and Algebraic Computation, pages 216-226, 1979. Google Scholar
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