Tight Bounds for Feedback Vertex Set Parameterized by Clique-width

TL;DR

This work advances the parameterized complexity of Feedback Vertex Set by introducing a novel acyclicity representation for labeled graphs and leveraging fast convolutions to count solutions modulo 2. It achieves a tight O(6^k n^c) time for FVS parameterized by clique-width, plus a one-sided Monte-Carlo decision algorithm, and proves SETH-based lower bounds matching these upper bounds. An independent treewidth-based counting algorithm in O*(3^tw) is also provided, along with a tight O*(18^cw) algorithm for Connected FVS via an isolating representative technique. Together, these results close several open gaps between counting, decision, and connectivity problems under structural parameters and establish new convolution techniques of broad potential.

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 $O(6^kn^c)$ 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 $O(3^kn^c)$ 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 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 $O(18^kn^c)$ decides the existence of a connected feedback vertex set of size $b$ in $G$. We provide a matching lower bound under SETH.

Figures (5)

• Figure 1: A $k$-labeled forest $F$ ($k=2$) and its corresponding pattern. Note that we upper bound the multiplicity of each label in each connected component by $2$, and the multiplicity of each connected component by $2$ as well. Removing additional unit-vectors happens in a later stage.
• Figure 2: The three patterns resulting from applying the reduction rule to the pattern $p$. We choose $x$ to be the zero vector as specified by the reduction rule. Note that the total sum of non-unit vectors different from the zero vector decreases from $3$ in $p$ to $2$, $0$ and $0$ in the three resulting patterns, respectively. Note also that we keep the upper bound $2$ on the sums in each index.
• Figure 3: A $k$-labeled forest $F$ ($k=2$), its corresponding pattern and the corresponding canonical forest. Note that we upper bound the multiplicity of each label in each connected component by $2$, and the multiplicity of each connected component by $2$ as well.
• Figure 4: On the left, the graph $G$ consisting of $n$ path sequences. We depict a path gadget as a square with its clique vertices drawn inside of it. Each column corresponds to a constraint with a constraint gadget hanging to it. The deletion edges between the first constraint gadget and the first column are drawn as dotted lines, where the rest are omitted for clarity. From the drawn gadgets, it can be seen that this graph $G$ corresponds to an instance where the first constraint admits two satisfying assignments, the second has four, and the third has two. The variables underlying the first constraint are the second and the last variables, as the deletion edges are drawn accordingly. On the right, we depict a path gadget. Normal edges are depicted as dashed lines, while deletion edges are depicted as solid lines. The deletion edges are drawn between the cycles $C_i$ and only the clique vertex $x_2$ for clarity. Note that a deletion edge to $v_2$ means that if the path gadget has state $2$ then $v_2$ must be in the solution (and hence, state ${\boldsymbol{O}}$), a deletion edge to $u_1$ means that $v_1$ is disconnected from $r$ inside the gadget (and hence, state ${\boldsymbol{D}}$), while a deletion edge to $w_3$ means that $v_3$ is connected to $r$ through $u_3$ (state ${\boldsymbol{C}}$).
• Figure 5: An entry/exit structure of a path gadget. The vertex $v_i$ is the boundary vertex.

Theorems & Definitions (109)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Remark 8
• Definition 9
• Definition 10
• Definition 11