凸回忆的有效逼近
Efficient Approximation of Convex Recolorings
摘要：
一棵树的着色是凸的，如果属于任何颜色的顶点诱导一个连通子树;一个部分着色（它将颜色分配给一些顶点）是凸的，如果它可以被完成为一个凸（全）着色。树的凸着色出现在遗传学、语言学等领域。例如，完美的系统发育树是其中每个特征的状态诱导树的凸着色的树。对于完全凸树的研究通常集中在寻找一棵树，使得其顶点的预定部分着色很少是凸的。当一棵树的着色不是凸的时，人们希望知道它离凸着色有多远。在[19]中，定义了一个自然的距离度量，称为包围距离：使着色凸所需的顶点处的颜色变化的最小数量。这可以被看作是最小化“例外顶点”的数量。最接近的凸着色。证明了该问题是NP-难的，即使对色弦也是如此。本文继续文[19]的工作，给出了一个运行时间为O（cn）的字符串凸排图的2-近似算法，其中c是颜色数，n是输入的大小;给出了一个运行时间为O（cn ^2）的树凸排图的3-近似算法.
英文摘要：
A coloring of a tree is convex if the vertices that pertain to any color induce a connected subtree; a partial coloring (which assigns colors to some of the vertices) is convex if it can be completed to a convex (total) coloring. Convex coloring of trees arise in areas such as phylogenetics, linguistics, etc. eg, a perfect phylogenetic tree is one in which the states of each character induce a convex coloring of the tree. Research on perfect phylogeny is usually focused on finding a tree so that few predetermined partial colorings of its vertices are convex. When a coloring of a tree is not convex, it is desirable to know "how far" it is from a convex one. In [19], a natural measure for this distance, called the recoloring distance was defined: the minimal number of color changes at the vertices needed to make the coloring convex. This can be viewed as minimizing the number of "exceptional vertices" w.r.t. to a closest convex coloring. The problem was proved to be NP-hard even for colored string. In this paper we continue the work of [19], and present a 2-approximation algorithm of convex recoloring of strings whose running time O(cn), where c is the number of colors and n is the size of the input, and an O(cn^2)-time 3-approximation algorithm for convex recoloring of trees.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
