Tight Bounds for Feedback Vertex Set Parameterized by Clique-width

Abstract: We introduce a new notion of acyclicity representation in labeled graphs, and present three applications thereof. Our main result is an algorithm that, given a graph $G$ and a $k$-clique expression of $G$, in time $O(6<sup>kn<sup>c)$ counts modulo $2$ the number of feedback vertex sets of $G$ of each size. We achieve this through an involved subroutine for merging partial solutions at union nodes in the expression. In the usual way this results in a one-sided error Monte-Carlo algorithm for solving the decision problem in the same time. We complement these by a matching lower bound under the Strong Exponential-Time Hypothesis (SETH). This closes an open question that appeared multiple times in the literature [ESA 23, ICALP 24, IPEC 25]. We also present an algorithm that, given a graph $G$ and a tree decomposition of width $k$ of $G$, in time $O(3<sup>kn<sup>c)$ counts modulo $2$ the number of feedback vertex sets of $G$ of each size. This matches the known SETH-tight bound for the decision version, which was obtained using the celebrated cut-and-count technique [FOCS 11, TALG 22]. Unlike other applications of cut-and-count, which use the isolation lemma to reduce a decision problem to counting solutions modulo $2$, this bound was obtained via counting other objects, leaving the complexity of counting solutions modulo $2$ open. Finally, we present a one-sided error Monte-Carlo algorithm that, given a graph $G$ and a $k$-clique expression of $G$, in time $O(18<sup>kn<sup>c)$ decides the existence of a connected feedback vertex set of size $b$ in $G$. We provide a matching lower bound under SETH.