部分可着色图的近似算法
Approximation Algorithms for Partially Colorable Graphs
摘要：
图的着色问题是算法理论研究的一个中心课题。研究了部分可着色图的部分着色问题。对于$α\leq 1$和$k \in \mathbb{Z}^+$，我们说一个图$G=(V,E)$是$α$ -部分$k$ -可着色的，如果存在基数$ |S | \geq α| V |$的子集$S\subset V$，使得在$S$上导出的图是$k$ -可着色的。部分$k$ -可着色性是一个比$k$ -可着色性更鲁棒的图的结构性质.对于在实践中出现的图，部分$k$ -可着色性可能是比$k$ -可着色性更好的概念，因为在实践中出现的数据通常包含各种形式的噪声。我们给出了一个多项式时间算法，作为输入的$(1 - ε)$ -部分$3$ -可着色图$G$和一个常数$γ\in [ε, 1/10]$，并使用$\tilde{O}\left(n^{0.25 + O(γ^{1/2})} \right)$颜色的$(1 - ε/γ)$分数的顶点。我们还研究了部分$3$ -可着色图和部分$2$ -可着色图的自然半随机实例族，并给出了这些实例族的更强的双准则逼近保证。
英文摘要：
Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $α\leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is $α$-partially $k$-colorable, if there exists a subset $S\subset V$ of cardinality $ |S | \geq α| V |$ such that the graph induced on $S$ is $k$-colorable. Partial $k$-colorability is a more robust structural property of a graph than $k$-colorability. For graphs that arise in practice, partial $k$-colorability might be a better notion to use than $k$-colorability, since data arising in practice often contains various forms of noise. We give a polynomial time algorithm that takes as input a $(1 - ε)$-partially $3$-colorable graph $G$ and a constant $γ\in [ε, 1/10]$, and colors a $(1 - ε/γ)$ fraction of the vertices using $\tilde{O}\left(n^{0.25 + O(γ^{1/2})} \right)$ colors. We also study natural semi-random families of instances of partially $3$-colorable graphs and partially $2$-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances.
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