用Treewidth参数化的平面小分支的复杂度二分法

Julien Baste, Ignasi Sau, D. Thilikos
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引用次数: 4

摘要

对于固定图H,我们感兴趣的是以下问题的参数化复杂度,称为{H}-M-Deletion,由输入图的树宽tw参数化:给定一个n顶点图G和一个整数k,判断是否存在S subseteq V(G),且|S| <= k,使得G set- S不包含H作为子集。在之前的工作[IPEC, 2017]中,我们证明了如果H是平面且连通的,那么在ETH下,问题不能在2^{o(tw)} * n^{o(1)}时间内解决,而可以在2^{o(tw * log tw)} * n^{o(1)}时间内解决。在本文中,我们设法对{H}-M-Deletion的最优渐近复杂度进行了分类,当H是一个最多有5个顶点的连通平面图时。在29种可能性中(抛弃H = K_1的平凡情况),我们证明了其中9种在2^{Theta (tw)} * n^{O(1)}时间内可解,另外20种在2^{Theta (tw * log tw)} * n^{O(1)}时间内可解。也就是说,我们证明了K_4和菱形图是在2^{Theta (tw * log tw)} * n^{O(1)}时间内问题在最多4个顶点上可解的图,而椅子和横幅是在5个顶点上问题在2^{Theta (tw)} * n^{O(1)}时间内可解的图。对于H作为拓扑次元被禁止的问题版本,H = K_{1,4}的情况可以在2^{Theta (tw)} * n^{O(1)}时间内求解。据我们所知,这显示了两个问题的计算复杂度之间的第一个区别。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A Complexity Dichotomy for Hitting Small Planar Minors Parameterized by Treewidth
For a fixed graph H, we are interested in the parameterized complexity of the following problem, called {H}-M-Deletion, parameterized by the treewidth tw of the input graph: given an n-vertex graph G and an integer k, decide whether there exists S subseteq V(G) with |S| <= k such that G setminus S does not contain H as a minor. In previous work [IPEC, 2017] we proved that if H is planar and connected, then the problem cannot be solved in time 2^{o(tw)} * n^{O(1)} under the ETH, and can be solved in time 2^{O(tw * log tw)} * n^{O(1)}. In this article we manage to classify the optimal asymptotic complexity of {H}-M-Deletion when H is a connected planar graph on at most 5 vertices. Out of the 29 possibilities (discarding the trivial case H = K_1), we prove that 9 of them are solvable in time 2^{Theta (tw)} * n^{O(1)}, and that the other 20 ones are solvable in time 2^{Theta (tw * log tw)} * n^{O(1)}. Namely, we prove that K_4 and the diamond are the only graphs on at most 4 vertices for which the problem is solvable in time 2^{Theta (tw * log tw)} * n^{O(1)}, and that the chair and the banner are the only graphs on 5 vertices for which the problem is solvable in time 2^{Theta (tw)} * n^{O(1)}. For the version of the problem where H is forbidden as a topological minor, the case H = K_{1,4} can be solved in time 2^{Theta (tw)} * n^{O(1)}. This exhibits, to the best of our knowledge, the first difference between the computational complexity of both problems.
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