Linear arrangements of graphs are a well-known type of graph labeling and are found at the heart of many important computational problems, such as the Minimum Linear Arrangement Problem (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 typically drawn as semicircles above the real line. In this paper we study the Maximum Linear Arrangement problem (MaxLA), the maximization variant of minLA and a less studied problem than minLA. We a devise new characterization of maximum arrangements of general graphs, and prove that MaxLA can be solved for cycle graphs in constant time, and for $k$-linear trees ($k\le2$) in time $O(n)$. We present a simple algorithm that solves a constrained variant of MaxLA, which we call bipartite MaxLA, in time $O(n)$. This algorithm has two promising characteristics. First, it solves MaxLA for most trees consisting of a few tenths of nodes. Second, it produces a high quality approximation to MaxLA for trees where the algorithm fails to solve MaxLA. Furthermore, we conjecture this algorithm solves MaxLA for at least $50\%$ of all free trees.

翻译：图的线性排列是一种众所周知的图标记方式，是许多重要计算问题的核心，例如最小线性排列问题（minLA）。线性排列通常定义为图 $n$ 个顶点的一种排列。一个直观的几何设定是顶点位于实数轴上从1开始的连续整数位置；边通常绘制为实数轴上方的半圆。本文研究最大线性排列问题（MaxLA），即minLA的最大化变体，且该问题比minLA研究较少。我们推导出一般图最大排列的新表征，并证明MaxLA可在常数时间内解决循环图，并在 $O(n)$ 时间内解决 $k$ 线性树（$k\le2$）。我们提出一种简单算法，可在 $O(n)$ 时间内解决MaxLA的一个约束变体，称为二分MaxLA。该算法具有两个有前景的特性。其一，它能解决由数十个节点组成的大部分树的MaxLA。其二，对于该算法无法解决MaxLA的树，它能生成MaxLA的高质量近似解。此外，我们推测该算法能解决至少 $50\%$ 自由树的MaxLA。