Graph Square Roots of Small Distance from Degree One Graphs

被引:0
|
作者
Petr A. Golovach
Paloma T. Lima
Charis Papadopoulos
机构
[1] University of Bergen,Department of Computer Science
[2] IT University of Copenhagen,Department of Computer Science
[3] University of Ioannina,Department of Mathematics
来源
Theory of Computing Systems | 2022年 / 66卷
关键词
Square root; Vertex cover; Structural parameterization;
D O I
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中图分类号
学科分类号
摘要
Given a graph class H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathscr{H}}$\end{document}, the task of the H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathscr{H}}$\end{document}-Square Root problem is to decide whether an input graph G has a square root H from H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathscr{H}}$\end{document}. We are interested in the parameterized complexity of the problem for classes H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathscr{H}}$\end{document} that are composed by the graphs at vertex deletion distance at most k from graphs of maximum degree at most one, that is, we are looking for a square root H such that there is a modulator S of size k such that H − S is the disjoint union of isolated vertices and disjoint edges. We show that different variants of the problems with constraints on the number of isolated vertices and edges in H − S are FPT when parameterized by k by demonstrating algorithms with running time 22O(k)⋅n5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2^{2^{\mathcal {O}(k)}}\cdot n^{5}$\end{document}. We further show that the running time of our algorithms is asymptotically optimal and it is unlikely that the double-exponential dependence on k could be avoided. In particular, we prove that the VC-kRoot problem, that asks whether an input graph has a square root with vertex cover of size at most k, cannot be solved in time 22o(k)⋅nO(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2^{2^{o(k)}}\cdot n^{\mathcal {O}(1)}$\end{document} unless the Exponential Time Hypothesis fails. Moreover, we point out that VC-kRoot parameterized by k does not admit a subexponential kernel unless P = NP.
引用
收藏
页码:821 / 846
页数:25
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