固有图与单位区间图的同时表示

IF 0.9 4区 计算机科学 Q4 COMPUTER SCIENCE, SOFTWARE ENGINEERING
Ignaz Rutter, Darren Strash, Peter Stumpf, Michael Vollmer
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引用次数: 0

摘要

在组合学和几何的融合中,同时表示提供了一种实现具有共同结构的组合对象的方法。同时表征研究中的一个标准案例是向日葵案例,其中所有对象都具有相同的公共结构。虽然一般同时区间图的识别问题是np完全的,但三个或更多同时区间图的向日葵情况的复杂性目前是开放的。在本文中,我们解决了固有区间图的这个问题。在允许任意数量的同时图的情况下,我们给出了一个在线性时间内识别同时固有间隔图的算法。同时单位间隔图更加“刚性”,因此在表示上的自由度更小。我们证明它们可以在时间上被识别\(\mathcal {O}(|V|\cdot |E|)\)对于任意数量的同时图在向日葵的情况下,其中\(G=(V,E)\)是同时图的并集。我们进一步证明,如果同时图的数量不固定,这两个识别问题在一般情况下都是np完全的。在这个意义上,对向日葵案例的限制是必要的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Simultaneous Representation of Proper and Unit Interval Graphs

In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where all objects share the same common structure. While the recognition problem for general simultaneous interval graphs—the simultaneous version of arguably one of the most well-studied graph classes—is NP-complete, the complexity of the sunflower case for three or more simultaneous interval graphs is currently open. In this work we settle this question for proper interval graphs. We give an algorithm to recognize simultaneous proper interval graphs in linear time in the sunflower case where we allow any number of simultaneous graphs. Simultaneous unit interval graphs are much more ‘rigid’ and therefore have less freedom in their representation. We show they can be recognized in time \(\mathcal {O}(|V|\cdot |E|)\) for any number of simultaneous graphs in the sunflower case where \(G=(V,E)\) is the union of the simultaneous graphs. We further show that both recognition problems are in general NP-complete if the number of simultaneous graphs is not fixed. The restriction to the sunflower case is in this sense necessary.

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来源期刊
Algorithmica
Algorithmica 工程技术-计算机:软件工程
CiteScore
2.80
自引率
9.10%
发文量
158
审稿时长
12 months
期刊介绍: Algorithmica is an international journal which publishes theoretical papers on algorithms that address problems arising in practical areas, and experimental papers of general appeal for practical importance or techniques. The development of algorithms is an integral part of computer science. The increasing complexity and scope of computer applications makes the design of efficient algorithms essential. Algorithmica covers algorithms in applied areas such as: VLSI, distributed computing, parallel processing, automated design, robotics, graphics, data base design, software tools, as well as algorithms in fundamental areas such as sorting, searching, data structures, computational geometry, and linear programming. In addition, the journal features two special sections: Application Experience, presenting findings obtained from applications of theoretical results to practical situations, and Problems, offering short papers presenting problems on selected topics of computer science.
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