在有界树宽图上打小调。一种最优算法

Julien Baste, Ignasi Sau, D. Thilikos
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引用次数: 1

摘要

对于一个固定的有限的图集合F, F- m - deletion问题的问题是,给定一个n顶点的输入图G,求与F中图G中所有次要模型相交的最小顶点数。根据Courcelle定理,这个问题可以在fF (tw)·nO(1)时间内得到解,其中tw是G的树宽度,对于某个函数fF依赖于F。在最近的一系列文章中,我们提出了函数fF的渐近优化方案。在这里,我们提供了一个算法,表明fF (tw) = 2O(tw·log tw)对于每个集合F。在此之前,最著名的函数fF是在tw中的双指数函数。特别是,我们的算法极大地扩展了Jansen等人[SODA 2014]对于特定情况F = {K5, K3,3}的结果以及Kociumaka和Pilipczuk [Algorithmica 2019]对于有界属图的结果,并回答了Cygan等人提出的一个开放问题[Inf Comput 2017]。我们结合了几种成分,如动态规划中有界图的机制,通过代表,平壁定理,二维性,无关顶点技术,树宽调制器和突出替换。结合我们之前为特定集合F[理论计算机科学2020]和一般下界[J计算机系统科学2020]提供的单指数算法的结果,我们的算法产生以下复杂度二分法,当F = {H}包含单个连通图H时,假设指数时间假设:如果H是椅子或横幅的收缩,fH(tw) = 2Θ(tw),否则fH(tw) = 2Θ(tw·log tw)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Hitting Minors on Bounded Treewidth Graphs. IV. An Optimal Algorithm
For a fixed finite collection of graphs F , the F-M-Deletion problem asks, given an n-vertex input graph G, for the minimum number of vertices that intersect all minor models in G of the graphs in F . by Courcelle’s Theorem, this problem can be solved in time fF (tw) · nO(1), where tw is the treewidth of G, for some function fF depending on F . In a recent series of articles, we have initiated the programme of optimizing asymptotically the function fF . Here we provide an algorithm showing that fF (tw) = 2O(tw · log tw) for every collection F . Prior to this work, the best known function fF was double-exponential in tw . In particular, our algorithm vastly extends the results of Jansen et al. [SODA 2014] for the particular case F = {K5, K3,3} and of Kociumaka and Pilipczuk [Algorithmica 2019] for graphs of bounded genus, and answers an open problem posed by Cygan et al. [Inf Comput 2017]. We combine several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, Bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement. Together with our previous results providing single-exponential algorithms for particular collections F [Theor Comput Sci 2020] and general lower bounds [J Comput Syst Sci 2020], our algorithm yields the following complexity dichotomy when F = {H} contains a single connected graph H, assuming the Exponential Time Hypothesis: fH(tw) = 2Θ(tw) if H is a contraction of the chair or the banner, and fH(tw) = 2Θ(tw · log tw) otherwise.
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