We introduce determinantal sieving, a new, remarkably powerful tool in the toolbox of algebraic FPT algorithms. Given a polynomial $P(X)$ on a set of variables $X=\{x_1,\ldots,x_n\}$ and a linear matroid $M=(X,\mathcal{I})$ of rank $k$, both over a field $\mathbb{F}$ of characteristic 2, in $2^k$ evaluations we can sieve for those terms in the monomial expansion of $P$ which are multilinear and whose support is a basis for $M$. Alternatively, using $2^k$ evaluations of $P$ we can sieve for those monomials whose odd support spans $M$. Applying this framework, we improve on a range of algebraic FPT algorithms, such as: 1. Solving $q$-Matroid Intersection in time $O^*(2^{(q-2)k})$ and $q$-Matroid Parity in time $O^*(2^{qk})$, improving on $O^*(4^{qk})$ (Brand and Pratt, ICALP 2021) 2. $T$-Cycle, Colourful $(s,t)$-Path, Colourful $(S,T)$-Linkage in undirected graphs, and the more general Rank $k$ $(S,T)$-Linkage problem, all in $O^*(2^k)$ time, improving on $O^*(2^{k+|S|})$ respectively $O^*(2^{|S|+O(k^2 \log(k+|\mathbb{F}|))})$ (Fomin et al., SODA 2023) 3. Many instances of the Diverse X paradigm, finding a collection of $r$ solutions to a problem with a minimum mutual distance of $d$ in time $O^*(2^{r(r-1)d/2})$, improving solutions for $k$-Distinct Branchings from time $2^{O(k \log k)}$ to $O^*(2^k)$ (Bang-Jensen et al., ESA 2021), and for Diverse Perfect Matchings from $O^*(2^{2^{O(rd)}})$ to $O^*(2^{r^2d/2})$ (Fomin et al., STACS 2021) All matroids are assumed to be represented over a field of characteristic 2. Over general fields, we achieve similar results at the cost of using exponential space by working over the exterior algebra. For a class of arithmetic circuits we call strongly monotone, this is even achieved without any loss of running time. However, the odd support sieving result appears to be specific to working over characteristic 2.

翻译：我们提出行列筛法（ determinantal sieving），这是代数FPT算法工具箱中一种新颖且威力强大的工具。给定变量集 $X=\{x_1,\ldots,x_n\}$ 上的多项式 $P(X)$ 和秩为 $k$ 的线性拟阵 $M=(X,\mathcal{I})$，两者均定义于特征为2的域 $\mathbb{F}$ 上，通过 $2^k$ 次求值即可筛出 $P$ 单项式展开中所有多线性且支撑集为 $M$ 基的项。或者，利用 $2^k$ 次 $P$ 求值可筛出所有奇支撑张成 $M$ 的单项式。应用该框架，我们改进了系列代数FPT算法，例如： 1. 在 $O^*(2^{(q-2)k})$ 时间内求解 $q$-拟阵交问题，在 $O^*(2^{qk})$ 时间内求解 $q$-拟阵奇偶性问题，改进了 $O^*(4^{qk})$ 的结果（Brand 与 Pratt，ICALP 2021）； 2. 无向图中的 $T$-环、多彩 $(s,t)$-路径、多彩 $(S,T)$-链接问题，以及更一般的秩 $k$ $(S,T)$-链接问题，均在 $O^*(2^k)$ 时间内解决，改进了 $O^*(2^{k+|S|})$ 或 $O^*(2^{|S|+O(k^2 \log(k+|\mathbb{F}|))})$ 的结果（Fomin 等，SODA 2023）； 3. 多样X范式（Diverse X paradigm）的众多实例：在 $O^*(2^{r(r-1)d/2})$ 时间内寻找问题中具有最小互距离 $d$ 的 $r$ 个解，将 $k$-相异分支问题的求解时间从 $2^{O(k \log k)}$ 改进至 $O^*(2^k)$（Bang-Jensen 等，ESA 2021），并将多样完美匹配问题的求解时间从 $O^*(2^{2^{O(rd)}})$ 改进至 $O^*(2^{r^2d/2})$（Fomin 等，STACS 2021）。 所有拟阵均假设表示于特征为2的域上。对于一般域，通过在体外代数（exterior algebra）上运算，我们以指数空间为代价实现了类似结果。对于称为强单调（strongly monotone）的一类算术电路，甚至可在不损失运行时间的前提下实现该结果。然而，奇支撑筛法的结论似乎仅适用于特征为2的域。