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

TL;DR

This paper investigates the ideal membership problem for univariate ideals, providing algorithms for polynomial evaluation, vertex cover, and permanent computation based on rank and degree parameters, and establishing hardness results.

Contribution

It introduces new algorithms for polynomial evaluation and permanent computation in low-rank settings, and proves hardness of membership testing parameterized by the number of generators.

Findings

Efficient evaluation of remainders modulo univariate ideals for low-rank polynomials.

Polynomial-time algorithms for minimum vertex cover and permanent in low-rank adjacency matrices.

Hardness results for membership testing parameterized by the number of generators.

Abstract

Let $\mathbb{F}[X]$ be the polynomial ring over the variables $X=\{x_1,x_2, \ldots, x_n\}$. An ideal $I=\langle p_1(x_1), \ldots, p_n(x_n)\rangle$ generated by univariate polynomials $\{p_i(x_i)\}_{i=1}^n$ is a \emph{univariate ideal}. We study the ideal membership problem for the univariate ideals and show the following results. \item Let $f(X)\in\mathbb{F}[\ell_1, \ldots, \ell_r]$ be a (low rank) polynomial given by an arithmetic circuit where $\ell_i : 1\leq i\leq r$ are linear forms, and $I=\langle p_1(x_1), \ldots, p_n(x_n)\rangle$ be a univariate ideal. Given $\vec{\alpha}\in {\mathbb{F}}^n$, the (unique) remainder $f(X) \pmod I$ can be evaluated at $\vec{\alpha}$ in deterministic time $d^{O(r)}\cdot poly(n)$, where $d=\max\{\deg(f),\deg(p_1)\ldots,\deg(p_n)\}$. This yields an $n^{O(r)}$ 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 $n\times n$ matrix of rank $r$, over any field $\mathbb{F}$. Over $\mathbb{Q}$, an algorithm of similar run time for low rank permanent is due to Barvinok[Bar96] via a different technique. \item Let $f(X)\in\mathbb{F}[X]$ be given by an arithmetic circuit of degree $k$ ($k$ treated as fixed parameter) and $I=\langle p_1(x_1), \ldots, p_n(x_n)\rangle$. We show in the special case when $I=\langle x_1^{e_1}, \ldots, x_n^{e_n}\rangle$, we obtain a randomized $O^*(4.08^k)$ algorithm that uses $poly(n,k)$ space. \item Given $f(X)\in\mathbb{F}[X]$ by an arithmetic circuit and $I=\langle p_1(x_1), \ldots, p_k(x_k) \rangle$, membership testing is $W[1]$-hard, parameterized by $k$. The problem is $MINI[1]$-hard in the special case when $I=\langle x_1^{e_1}, \ldots, x_k^{e_k}\rangle$.