当前位置: X-MOL 学术arXiv.cs.CC › 论文详情
Maximum Clique in Disk-Like Intersection Graphs
arXiv - CS - Computational Complexity Pub Date : 2020-03-05 , DOI: arxiv-2003.02583
Édouard Bonnet; Nicolas Grelier; Tillmann Miltzow

We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark '90, Raghavan and Spinrad '03] to translates of any fixed convex set. We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. '18, Bonamy et al. '18] to homothets of a fixed centrally symmetric convex set. The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-hard. We observe that, so far, all the hardness proofs for Maximum Clique in intersection graph classes $\mathcal I$ follow the same road. They show that, for every graph $G$ of a large-enough class $\mathcal C$, the complement of an even subdivision of $G$ belongs to the intersection class $\mathcal I$. Then they conclude invoking the hardness of Maximum Independent Set on the class $\mathcal C$, and the fact that the even subdivision preserves that hardness. However there is a strong evidence that this approach cannot work for disk graphs [Bonnet et al. '18]. We suggest a new approach, based on a problem that we dub Max Interval Permutation Avoidance, which we prove unlikely to have a subexponential-time approximation scheme. We transfer that hardness to Maximum Clique in intersection graphs of objects which can be either half-planes (or unit disks) or axis-parallel rectangles. That problem is not amenable to the previous approach. We hope that a scaled down (merely NP-hard) variant of Max Interval Permutation Avoidance could help making progress on the disk case, for instance by showing the NP-hardness for (convex) pseudo-disks.
更新日期:2020-03-06

 

全部期刊列表>>
智控未来
聚焦商业经济政治法律
跟Nature、Science文章学绘图
控制与机器人
招募海内外科研人才,上自然官网
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
x-mol收录
湖南大学化学化工学院刘松
上海有机所
李旸
南方科技大学
西湖大学
X-MOL
支志明
中山大学化学工程与技术学院
试剂库存
天合科研
down
wechat
bug