Subset Feedback Vertex Set in Chordal and Split Graphs

In the Subset Feedback Vertex Set (Subset-FVS) problem the input is a graph G on n vertices, a subset T of vertices of G called the “terminal” vertices, and an integer k. The task is to determine whether there exists a subset of vertices of cardinality at most k which together intersect all cycles which pass through the terminals. Subset-FVS generalizes several well studied problems including Feedback Vertex Set and Multiway Cut. This problem is known to be NP-Complete, even in split graphs. Cygan et al. (SIAM J Discrete Math 27(1):290–309, 2013) proved that Subset-FVS is fixed parameter tractable (FPT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {FPT}$$\end{document}) in general graphs when parameterized by k. In split graphs a simple observation reduces the problem to an equivalent instance of the 3-Hitting Set problem with the same solution size. This directly implies, for Subset-FVSrestricted to split graphs, (i) an FPT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {FPT}$$\end{document} algorithm which solves the problem in O⋆(2.076k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}^{\star } (2.076^k)$$\end{document} time (The O⋆()\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}^{\star } ()$$\end{document} notation hides polynomial factors.) (Wahlström in Algorithms, measures and upper bounds for satisfiability and related problems. Ph.D. Thesis, Department of Computer and Information Science, Linköpings universitet, 2007), and (ii) a kernel of size O(k3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(k^3)$$\end{document}. We improve both these results for Subset-FVS on split graphs; we derive (i) a kernel of size O(k2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(k^2)$$\end{document} which is the best possible unless NP⊆coNP/poly\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf {NP}\subseteq {\mathsf {coNP}}/{\textsf {poly}}$$\end{document}, and (ii) an algorithm which solves the problem in time O∗(2k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}^*(2^k)$$\end{document}. Our algorithm, in fact, solves Subset-FVS on the more general class of chordal graphs, also in O∗(2k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}^*(2^k)$$\end{document} time. To the best of our knowledge, the fastest known exact algorithm for Subset-FVS on chordal graphs is based on the 3-Hitting Set algorithm of Fomin et al. (JACM 66(2):8, 2019) which runs in O∗(1.5182n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}^*(1.5182^n)$$\end{document} time. Applying the results of Fomin et al. (2019) to our FPT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {FPT}$$\end{document} algorithm yields two exact exponential-time algorithms for Subset-FVS on chordal graphs: a randomized algorithm which runs in O∗(1.5n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}^*(1.5^{n})$$\end{document} time, and a deterministic algorithm which runs in O∗((1.5+ε)n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}^*((1.5+\varepsilon )^{n})$$\end{document} time for any fixed ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document}.