Tight Bounds for Feedback Vertex Set Parameterized by Clique-width
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912740977672192 |
|---|---|
| author | Bojikian, Narek Kratsch, Stefan |
| author_facet | Bojikian, Narek Kratsch, Stefan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01900 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tight Bounds for Feedback Vertex Set Parameterized by Clique-width Bojikian, Narek Kratsch, Stefan Data Structures and Algorithms 05C85 F.2 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. |
| title | Tight Bounds for Feedback Vertex Set Parameterized by Clique-width |
| topic | Data Structures and Algorithms 05C85 F.2 |
| url | https://arxiv.org/abs/2512.01900 |