Determinantal Sieving

TL;DR

This work introduces determinantal sieving, a novel algebraic technique that extends multilinear detection to sieve for monomials whose supports form a basis (or span) a given linear matroid over fields of characteristic 2. It achieves $O^*(2^k)$-time randomized algorithms (with polynomial space) for a broad class of problems by evaluating a polynomial on carefully chosen evaluation points and using determinant (or exterior-algebra) encodings to filter the target monomials; over general fields, exterior algebra enables similar results at higher cost. The authors apply the framework to matroid covering/packing/intersection problems, diverse solution collections, and path/linkage/subgraph problems, delivering faster runtimes (e.g., $q$-Matroid Intersection in $O^*(2^{(q-2)k})$ for linear matroids in characteristic 2) and faster long-path/long-cycle algorithms (e.g., $O^*(1.66^k)$ for Long $(s,t)$-Path). The approach also unifies and simplifies previous algebraic FPT methods, reduces space to polynomial, and provides flexible templates for balancing constraints via matroid structures. This framework broadens the reach of algebraic-FPT techniques and promises practical implications for a wide array of combinatorial problems where matroid-based constraints arise.

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. Long $(s,t)$-Path in $O^*(1.66^k)$ time, improving on $O^*(2^k)$, and Rank $k$ $(S,T)$-Linkage in so-called frameworks in $O^*(2^k)$ time, improving on $O^*(2^{|S|+O(k^2 \log(k+|\mathbb{F}|))})$ over general fields (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.

Theorems & Definitions (67)

• Lemma 1.1: Multilinear detection Bjorklund14detsumBjorklundHKK17narrow
• Lemma 1.2: Constrained multilinear detection BjorklundKK16
• Theorem 1.3: Basis sieving
• Theorem 1.4: Odd sieving
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• corollary 1.8
• Theorem 1.9
• Theorem 1.10