Linear arrangements of graphs are a well-known type of graph labeling and are found in many important computational problems, such as the Minimum Linear Arrangement Problem ($\texttt{minLA}$). 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. In this paper we study the Maximum Linear Arrangement problem ($\texttt{MaxLA}$), the maximization variant of $\texttt{minLA}$. We devise a new characterization of maximum arrangements of general graphs, and prove that $\texttt{MaxLA}$ can be solved for cycle graphs in constant time, and for $k$-linear trees ($k\le2$) in time $O(n)$. We present two constrained variants of $\texttt{MaxLA}$ we call $\texttt{bipartite MaxLA}$ and $\texttt{1-thistle MaxLA}$. We prove that the former can be solved in time $O(n)$ for any bipartite graph; the latter, by an algorithm that typically runs in time $O(n^4)$ on unlabelled trees. The combination of the two variants has two promising characteristics. First, it solves $\texttt{MaxLA}$ for almost all trees consisting of a few tenths of nodes. Second, we prove that it constitutes a $3/2$-approximation algorithm for $\texttt{MaxLA}$ for trees. Furthermore, we conjecture that $\texttt{bipartite MaxLA}$ solves $\texttt{MaxLA}$ for at least $50\%$ of all free trees.

翻译：图的线性排列是一类著名的图标记方式，出现在许多重要计算问题中，例如最小线性排列问题（$\texttt{minLA}$）。线性排列通常定义为图$n$个顶点的一个排列。一个直观的几何设置是顶点位于实轴上从1开始的连续整数位置；边常被绘制为实轴上方的半圆。本文研究最大线性排列问题（$\texttt{MaxLA}$），即$\texttt{minLA}$的最大化变体。我们提出了一般图最大排列的新刻画，并证明$\texttt{MaxLA}$可在常数时间内求解环图，对于$k$-线性树（$k\le2$）可在$O(n)$时间内求解。我们引入了$\texttt{MaxLA}$的两个约束变体，分别称为$\texttt{二分图MaxLA}$和$\texttt{1-thistle MaxLA}$。我们证明前者可在$O(n)$时间内求解任意二分图；后者可通过一种典型运行时间为$O(n^4)$的算法求解无标号树。这两个变体的结合具有两个有前景的特性：首先，它能够求解由几十个节点组成的几乎所有树的$\texttt{MaxLA}$；其次，我们证明它构成了树的$\texttt{MaxLA}$的一个$3/2$近似算法。此外，我们猜想$\texttt{二分图MaxLA}$能够求解至少$50\%$的自由树的$\texttt{MaxLA}$。