🤖 AI Summary
This paper addresses the monomial selection problem for polynomials under matroid constraints: efficiently identifying terms whose support forms a basis of a linear matroid—or whose odd support spans the matroid—over a field of characteristic two. We introduce the *determinant sieve*, the first method coupling determinant-based algebraic structure with matroid constraints, requiring only $2^k$ polynomial evaluations. This breaks the long-standing barriers of traditional FPT algorithms, which rely on $4^{qk}$-time or doubly exponential approaches. Our framework unifies and optimizes multiple fundamental problems—including $q$-matroid intersection, $T$-cycles, colorful paths, rank-$k$ linkages, and diverse solutions—reducing their time complexity to $O^*(2^{ck})$. Several results achieve optimal exponential bounds, significantly advancing the intersection of algebraic FPT techniques and matroid algorithms.
📝 Abstract
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 $O^*(4^{qk})$ over general fields (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 $O^*(2^{k+|S|})$ and $O^*(2^{|S|+O(k^2 log(k+|mathbb{F}|))})$ respectively (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) Here, all matroids are assumed to be represented over fields 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.
