最大线性排列：特定图类的精确算法和广泛图类的近似算法
Maximum Linear Arrangement: exact algorithms for specific classes of graphs and approximation algorithms for wide classes of graphs
AI总结：
本文研究最大线性排列问题，针对特定图类提出精确算法，针对广泛图类提出近似算法，证明了 bipartite MaxLA 对树的 3/2-近似性。
摘要：
图的线性排列是一种广为人知的图标记类型，广泛应用于重要计算问题。线性排列通常定义为图的顶点排列。直观的几何设定是顶点位于实线上连续整数位置，从1开始；边常画成实线上方的半圆。著名的计算问题是最小线性排列问题（minLA），目标是找到使边长总和最小的排列。本文研究最大线性排列问题（MaxLA），即minLA的对应问题。我们提出了一种新的最大排列一般图的特征化，并证明对于k-正则图（k≤2）可在O(n)时间内解决，对于k-线性树（k≤2）也可在O(n)时间内解决。我们提出了两个受约束的MaxLA变种，称为bipartite MaxLA和1-thistle MaxLA。我们证明前者可在O(n)时间内解决任何连通双色图；后者可通过通常在O(n³ log n)时间内运行的算法解决无标记树。我们证明bipartite MaxLA是MaxLA在树上的3/2近似算法。
英文摘要：
Linear arrangements of graphs are a well-known type of graph labeling and are found in many important computational problems. A linear arrangement is usually defined as a permutation of the $n$ vertices of a graph. An intuitive geometric setting is that of vertices lying on consecutive integer positions in the real line, starting at 1; edges are often drawn as semicircles above the real line. A well-known computational problem is the Minimum Linear Arrangement Problem (${\tt minLA}$) where the goal is to find an arrangement that minimizes the sum of edge lengths. In this paper we study the Maximum Linear Arrangement problem (${\tt MaxLA}$), the counterpart of ${\tt minLA}$. We devise a new characterization of maximum arrangements of general graphs, and prove that ${\tt MaxLA}$ can be solved for $k$-regular graphs ($k\le2$) in time $O(n)$, and for $k$-linear trees ($k\le2$) in time $O(n)$. We present two constrained variants of ${\tt MaxLA}$ we call ${\tt bipartite MaxLA}$ and ${\tt 1-thistle MaxLA}$. We prove that the former can be solved in time $O(n)$ for any connected bipartite graph; the latter can be solved by an algorithm that typically runs in time $O(n^3\log n)$ on unlabeled trees. We show that ${\tt bipartite MaxLA}$ is a $3/2$-approximation algorithm for ${\tt MaxLA}$ for trees.
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