当前位置: X-MOL 学术arXiv.cs.DM › 论文详情
Grundy Coloring & friends, Half-Graphs, Bicliques
arXiv - CS - Discrete Mathematics Pub Date : 2020-01-11 , DOI: arxiv-2001.03794
Pierre Aboulker; Édouard Bonnet; Eun Jung Kim; Florian Sikora

The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order $\sigma$, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering $\sigma$, i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force $f(k)n^{2^{k-1}}$-time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where $k$ is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on $k$ in the exponent of $n$ can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS '17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on $K_{t,t}$-free graphs for b-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest.
更新日期:2020-01-14

 

全部期刊列表>>
物理学研究前沿热点精选期刊推荐
chemistry
自然职位线上招聘会
欢迎报名注册2020量子在线大会
化学领域亟待解决的问题
材料学研究精选新
GIANT
ACS ES&T Engineering
ACS ES&T Water
ACS Publications填问卷
屿渡论文,编辑服务
阿拉丁试剂right
南昌大学
王辉
南方科技大学
彭小水
隐藏1h前已浏览文章
课题组网站
新版X-MOL期刊搜索和高级搜索功能介绍
ACS材料视界
天合科研
x-mol收录
赵延川
李霄羽
廖矿标
朱守非
试剂库存
down
wechat
bug