Tight Bounds for Feedback Vertex Set Parameterized by Clique-Width

Abstract

We introduce a new form 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^kn^c) counts modulo 2 the number of feedback vertex sets of G of each size. We achieve this through dynamic programming on the clique expression with 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], and significantly improves the dependence on k compared to the 𝒪(15^k 2^((ω+1)k) k^c n) time algorithm due to Bergougnoux and Kanté [TCS 2019] at the cost of being randomized.
We also present an algorithm that, given a graph G and a tree decomposition of width k of G, in time 𝒪(3^kn^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 open the complexity of counting solutions modulo 2.
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^k n^c) decides the existence of a connected feedback vertex set of size t in G. We provide a matching lower bound under SETH.