{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/linear-time-kernelization-for-feedback-vertex","title":"Linear-time Kernelization for Feedback Vertex Set","arxiv_id":"1608.01463","date":"2016-08-04","proceeding":null,"authors":["Yoichi Iwata"],"abstract":"In this paper, we propose an algorithm that, given an undirected graph $G$ of $m$ edges and an integer $k$, computes a graph $G'$ and an integer $k'$ in $O(k^4 m)$ time such that (1) the size of the graph $G'$ is $O(k^2)$, (2) $k'\\leq k$, and (3) $G$ has a feedback vertex set of size at most $k$ if and only if $G'$ has a feedback vertex set of size at most $k'$. This is the first linear-time polynomial-size kernel for Feedback Vertex Set. The size of our kernel is $2k^2+k$ vertices and $4k^2$ edges, which is smaller than the previous best of $4k^2$ vertices and $8k^2$ edges. Thus, we improve the size and the running time simultaneously. We note that under the assumption of $\\mathrm{NP}\\not\\subseteq\\mathrm{coNP}/\\mathrm{poly}$, Feedback Vertex Set does not admit an $O(k^{2-\\epsilon})$-size kernel for any $\\epsilon>0$. Our kernel exploits $k$-submodular relaxation, which is a recently developed technique for obtaining efficient FPT algorithms for various problems. The dual of $k$-submodular relaxation of Feedback Vertex Set can be seen as a half-integral variant of $A$-path packing, and to obtain the linear-time complexity, we propose an efficient augmenting-path algorithm for this problem. We believe that this combinatorial algorithm is of independent interest. A solver based on the proposed method won first place in the 1st Parameterized Algorithms and Computational Experiments (PACE) challenge.","url_abs":"http://arxiv.org/abs/1608.01463v3","url_pdf":"http://arxiv.org/pdf/1608.01463v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"linear-time-kernelization-for-feedback-vertex","repo_url":"https://github.com/wata-orz/fvs","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"linear-time-kernelization-for-feedback-vertex","repo_url":"https://bitbucket.org/marcin_pilipczuk/fvs-experiments","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}