多项式Hadamard积的快速精确算法
Fast Exact Algorithms Using Hadamard Product of Polynomials
摘要：
假设$C$是作为输入给出的$poly(n)$大小的算术电路，其计算多项式$f\in\mathbb{F}[X]$，其中$X=\{x_1,x_2,\ldots,x_n\}$和$\mathbb{F}$是其中可以有效地执行字段算术的任何字段。我们获得了新的算法，以下两个问题首先研究Koutis和威廉姆斯\cite{Kou08, Wi09, KW16}。k-MLC：计算多项式$f$中所有次数$k$多线性单项式的系数之和。k-MMD：测试在多项式$f$中是否存在非零次$k$多线性单项式。我们的算法是基于这样一个事实，即Hadamard产品$f\circ S_{n,k}$，是$f$，其中$S_{n,k}$是$k^{th}$基本对称多项式的程度$k$多线性部分。1.对于k-MLC问题，我们给出了一个运行时的确定性算法$O^*(n^{k/2+c\log k})$（其中$c$为常数），回答了Koutis和Williams \cite [ICALP'09]{KW 16}的一个公开问题.作为推论，我们给出了几个组合问题的$O^*(\binom{n}{\downarrow k/2})$ -时间精确计数算法：k-树，t-支配集，m-维k-匹配。2.对于k-MMD问题，我们给出了一个运行时的随机化算法$4.32^k\cdot poly(n,k)$。我们的算法只使用$poly(n,k)$空间。这与$k-MMD$的最新算法\cite{BDH18}的运行时间相匹配，该算法需要指数（在$k$中）空间。其他结果包括快速确定性算法的k-MLC和k-MMD问题的深度三个电路。
英文摘要：
Let $C$ be an arithmetic circuit of $poly(n)$ size given as input that computes a polynomial $f\in\mathbb{F}[X]$, where $X=\{x_1,x_2,\ldots,x_n\}$ and $\mathbb{F}$ is any field where the field arithmetic can be performed efficiently. We obtain new algorithms for the following two problems first studied by Koutis and Williams \cite{Kou08, Wi09, KW16}. k-MLC: Compute the sum of the coefficients of all degree-$k$ multilinear monomials in the polynomial $f$. k-MMD: Test if there is a nonzero degree-$k$ multilinear monomial in the polynomial $f$. Our algorithms are based on the fact that the Hadamard product $f\circ S_{n,k}$, is the degree-$k$ multilinear part of $f$, where $S_{n,k}$ is the $k^{th}$ elementary symmetric polynomial. 1. For k-MLC problem, we give a deterministic algorithm of run time $O^*(n^{k/2+c\log k})$ (where $c$ is a constant), answering an open question of Koutis and Williams \cite[ICALP'09]{KW16}. As corollaries, we show $O^*(\binom{n}{\downarrow k/2})$-time exact counting algorithms for several combinatorial problems: k-Tree, t-Dominating Set, m-Dimensional k-Matching. 2. For k-MMD problem, we give a randomized algorithm of run time $4.32^k\cdot poly(n,k)$. Our algorithm uses only $poly(n,k)$ space. This matches the run time of a recent algorithm \cite{BDH18} for $k-MMD$ which requires exponential (in $k$) space. Other results include fast deterministic algorithms for k-MLC and k-MMD problems for depth three circuits.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
