完全图的斜率约束图的np完备性

Q4 Mathematics

International Journal of Computational Geometry & Applications Pub Date : 2020-01-14 DOI:10.20382/jocg.v11i1a14

Cédric Pilatte

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引用次数: 0

摘要

我们证明了下面问题的np完备性。给定一组$S$的$n$斜率和一个整数$k\geq 1$，是否有可能在平面上的$k$顶点上画一个完整的图，只使用来自$S$的斜率?同样地，在一般位置上是否存在一个由$k$点组成的集合$K$，使得$K$两点之间的每一段的斜率都在$S$ ?然后我们提出了一个多项式算法当$n\leq 2k-c$，条件是R.E. Jamison的猜想。对于$n=k$, Wade和Chu在$\mathcal{O}(n^4)$中提出了一个算法。在这种情况下，我们的算法是线性的，不依赖于Jamison的猜想。

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NP-completeness of slope-constrained drawing of complete graphs

We prove the NP-completeness of the following problem. Given a set $S$ of $n$ slopes and an integer $k\geq 1$, is it possible to draw a complete graph on $k$ vertices in the plane using only slopes from $S$? Equivalently, does there exist a set $K$ of $k$ points in general position such that the slope of every segment between two points of $K$ is in $S$? We then present a polynomial algorithm for this question when $n\leq 2k-c$, conditional on a conjecture of R.E. Jamison. For $n=k$, an algorithm in $\mathcal{O}(n^4)$ was proposed by Wade and Chu. For this case, our algorithm is linear and does not rely on Jamison's conjecture.

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来源期刊

International Journal of Computational Geometry & Applications 数学-计算机：理论方法

CiteScore

0.80

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The International Journal of Computational Geometry & Applications (IJCGA) is a quarterly journal devoted to the field of computational geometry within the framework of design and analysis of algorithms. Emphasis is placed on the computational aspects of geometric problems that arise in various fields of science and engineering including computer-aided geometry design (CAGD), computer graphics, constructive solid geometry (CSG), operations research, pattern recognition, robotics, solid modelling, VLSI routing/layout, and others. Research contributions ranging from theoretical results in algorithm design — sequential or parallel, probabilistic or randomized algorithms — to applications in the above-mentioned areas are welcome. Research findings or experiences in the implementations of geometric algorithms, such as numerical stability, and papers with a geometric flavour related to algorithms or the application areas of computational geometry are also welcome.