有向图着色与到非环的距离

IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS
Ararat Harutyunyan, Michael Lampis, Nikolaos Melissinos
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引用次数: 0

摘要

在k-有向图着色中,我们给定一个有向图,并要求将其顶点划分为最多k个集合,使每个集合产生DAG。这个众所周知的问题是np困难的,因为它推广了(无向)k-着色,但如果输入的有向图是无环的,它就变得平凡了。这就提出了一个自然的参数化复杂性问题,即当输入“几乎”是无循环时将会发生什么。在本文中,我们使用测量输入在有向或无向意义上的距离的参数来研究这个问题。在有向意义上,到非环的距离最自然的概念可能是有向反馈顶点集。已知,对于所有k≥2的有向图,k-有向图着色在最大k + 4的有向反馈顶点集上是np困难的。我们加强了这个结果,证明对于所有k≥2,对于大小恰好为k的有向反馈顶点集,k-有向图着色已经是np困难的。这立即提供了一个二分类,因为如果有向反馈顶点集的大小不超过k−1,k-有向图着色是平凡的。改进我们的简化,我们得到了三个进一步的结果:(i) 2-有向图着色是np困难的有向反馈顶点集最多为3的有向图;(ii)对于所有k≥2,对于最大为k2的反馈弧集图,k-有向图着色是np困难的;有趣的是,这导致了第二个二分法,因为我们表明,如果反馈弧集的大小最多为k2−1,那么问题是FPT除以k;(iii)对于有向反馈顶点k的图,k-有向图着色是np困难的,即使最大度Δ不超过4k−1;我们证明了这也几乎是紧的,因为对于大小为k且Δ≤4k−3的有向反馈顶点集的有向图,问题变成了FPT。由于这些结果暗示问题在有界有向树宽度的图上也是np困难的,因此我们考虑测量与底层图的非环距离的参数。从积极的方面来看,我们证明了k-有向图着色允许一个由树宽度参数化的FPT算法,其参数依赖性为(tw!)ktw。由于这比(无向)k-着色的ktw依赖性要差得多,我们提出了tw!因素可以消除。我们在这部分的主要贡献是在否定的情况下解决这个问题,并表明我们的算法本质上是最优的,即使对于更受限制的参数树深和k = 2。具体地说,我们证明了一种求解具有tdo(td)依赖性的2-有向图着色的FPT算法将与ETH相矛盾。然后,我们考虑比赛的级别。我们知道,决定一场比赛是否有两种颜色是np完全的。我们提出了一个算法来决定我们是否可以在\(O^{*}({\sqrt [3]{6}}^{n})\)时间内进行2色比赛。最后,我们解释了如何修改该算法以确定比赛是否可k色。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Digraph Coloring and Distance to Acyclicity

Digraph Coloring and Distance to Acyclicity

In k-Digraph Coloring we are given a digraph and are asked to partition its vertices into at most k sets, so that each set induces a DAG. This well-known problem is NP-hard, as it generalizes (undirected) k-Coloring, but becomes trivial if the input digraph is acyclic. This poses the natural parameterized complexity question of what happens when the input is “almost” acyclic. In this paper we study this question using parameters that measure the input’s distance to acyclicity in either the directed or the undirected sense. In the directed sense perhaps the most natural notion of distance to acyclicity is directed feedback vertex set. It is already known that, for all k ≥ 2, k-Digraph Coloring is NP-hard on digraphs of directed feedback vertex set of size at most k + 4. We strengthen this result to show that, for all k ≥ 2, k-Digraph Coloring is already NP-hard for directed feedback vertex set of size exactly k. This immediately provides a dichotomy, as k-Digraph Coloring is trivial if directed feedback vertex set has size at most k − 1. Refining our reduction we obtain three further consequences: (i) 2-Digraph Coloring is NP-hard for oriented graphs of directed feedback vertex set at most 3; (ii) for all k ≥ 2, k-Digraph Coloring is NP-hard for graphs of feedback arc set of size at most k2; interestingly, this leads to a second dichotomy, as we show that the problem is FPT by k if feedback arc set has size at most k2 − 1; (iii) k-Digraph Coloring is NP-hard for graphs of directed feedback vertex k, even if the maximum degree Δ is at most 4k − 1; we show that this is also almost tight, as the problem becomes FPT for digraphs of directed feedback vertex set of size k and Δ ≤ 4k − 3. Since these results imply that the problem is also NP-hard on graphs of bounded directed treewidth, we then consider parameters that measure the distance from acyclicity of the underlying graph. On the positive side, we show that k-Digraph Coloring admits an FPT algorithm parameterized by treewidth, whose parameter dependence is (tw!)ktw. Since this is considerably worse than the ktw dependence of (undirected) k-Coloring, we pose the question of whether the tw! factor can be eliminated. Our main contribution in this part is to settle this question in the negative and show that our algorithm is essentially optimal, even for the much more restricted parameter treedepth and for k = 2. Specifically, we show that an FPT algorithm solving 2-Digraph Coloring with dependence tdo(td) would contradict the ETH. Then, we consider the class of tournaments. It is known that deciding whether a tournament is 2-colorable is NP-complete. We present an algorithm that decides if we can 2-color a tournament in \(O^{*}({\sqrt [3]{6}}^{n})\) time. Finally, we explain how this algorithm can be modified to decide if a tournament is k-colorable.

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来源期刊
Theory of Computing Systems
Theory of Computing Systems 工程技术-计算机:理论方法
CiteScore
1.90
自引率
0.00%
发文量
36
审稿时长
6-12 weeks
期刊介绍: TOCS is devoted to publishing original research from all areas of theoretical computer science, ranging from foundational areas such as computational complexity, to fundamental areas such as algorithms and data structures, to focused areas such as parallel and distributed algorithms and architectures.
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