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DOI: 10.4230/LIPIcs.FSTTCS.2018.7
URN: urn:nbn:de:0030-drops-99068
URL: http://drops.dagstuhl.de/opus/volltexte/2018/9906/
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Arvind, V. ; Chatterjee, Abhranil ; Datta, Rajit ; Mukhopadhyay, Partha

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

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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}>.

BibTeX - Entry

@InProceedings{arvind_et_al:LIPIcs:2018:9906,
  author =	{V. Arvind and Abhranil Chatterjee and Rajit Datta and Partha Mukhopadhyay},
  title =	{{Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators}},
  booktitle =	{38th IARCS Annual Conference on Foundations of Software  Technology and Theoretical Computer Science (FSTTCS 2018)},
  pages =	{7:1--7:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-093-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{122},
  editor =	{Sumit Ganguly and Paritosh Pandya},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9906},
  URN =		{urn:nbn:de:0030-drops-99068},
  doi =		{10.4230/LIPIcs.FSTTCS.2018.7},
  annote =	{Keywords: Combinatorial Nullstellensatz, Ideal Membership, Parametric Hardness, Low Rank Permanent}
}

Keywords: Combinatorial Nullstellensatz, Ideal Membership, Parametric Hardness, Low Rank Permanent
Seminar: 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)
Issue Date: 2018
Date of publication: 23.11.2018


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