超图的线性有序着色

IF 0.8 Q3 COMPUTER SCIENCE, THEORY & METHODS
Tamio-Vesa Nakajima, Stanislav Živný
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引用次数: 8

摘要

一个r-一致超图的线性有序(LO) k-着色赋值从{1,…, k}到每个顶点,这样,在每条边,(多)颜色集有一个唯一的最大值。同样地,对于r = 3,如果一条边的两个顶点被赋予相同的颜色,那么第三个顶点被赋予更大的颜色(而不是不同的颜色,就像在经典的非单色着色中一样)。Barto、Battistelli和Berg [STACS ' 21]在承诺约束满足问题(pcsp)的背景下研究了3-均匀超图上的LO着色。我们展示了两个结果。首先,给定一个允许LO - 2着色的3-均匀超图,我们可以在多项式时间内用\( k=O(\sqrt [3]{n \log \log n / \log n} \)找到LO - 2着色。其次,给定一个允许LO - 2着色的r-均匀超图,我们建立了对于每一个常数均匀性r≥k+2,寻找LO - 2着色的np -硬度。事实上,我们确定了所有均匀性r≥3的多态性仆从之间的关系,这揭示了r< k+2和r≥k+2之间的关键区别,这可能是独立的兴趣。利用pcsp的代数方法,我们实际上展示了一个更一般的结果,建立了寻找LO -着色的LO -可着色的r-均匀超图的np -硬度,对于2≤r≤k≤k和r≥k- r + 4。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Linearly Ordered Colourings of Hypergraphs
A linearly ordered (LO) k-colouring of an r-uniform hypergraph assigns an integer from {1, ... , k } to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for r = 3, if two vertices in an edge are assigned the same colour, then the third vertex is assigned a larger colour (as opposed to a different colour, as in classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS’21] studied LO colourings on 3-uniform hypergraphs in the context of promise constraint satisfaction problems (PCSPs). We show two results. First, given a 3-uniform hypergraph that admits an LO 2-colouring, one can find in polynomial time an LO k-colouring with \( k=O(\sqrt [3]{n \log \log n / \log n} \) . Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO k-colouring for every constant uniformity r≥k+2. In fact, we determine relationships between polymorphism minions for all uniformities r≥ 3, which reveals a key difference between r< k+2 and r≥ k+2 and which may be of independent interest. Using the algebraic approach to PCSPs, we actually show a more general result establishing NP-hardness of finding an LO k-colouring for LO ℓ-colourable r-uniform hypergraphs for 2 ≤ ℓ ≤ k and r ≥ k - ℓ + 4.
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来源期刊
ACM Transactions on Computation Theory
ACM Transactions on Computation Theory COMPUTER SCIENCE, THEORY & METHODS-
CiteScore
2.30
自引率
0.00%
发文量
10
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