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 ($\texttt{minLA}$). 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 $O(n)$ for any bipartite graph; the latter, by an algorithm that typically runs in $O(n^3)$ 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, it produces a high quality approximation to $\texttt{MaxLA}$ for trees where the algorithm fails to solve $\texttt{MaxLA}$. Furthermore, we conjecture that $\texttt{bipartite MaxLA}$ solves $\texttt{MaxLA}$ for at least $50\%$ of all free trees.

翻译：图的线性排列是一种众所周知的图标记方式，也是许多重要计算问题的核心，例如最小线性排列问题（$\texttt{minLA}$）。线性排列通常定义为图中$n$个顶点的一个排列。一个直观的几何设置是：顶点位于实数轴上从1开始的连续整数位置上，而边通常被绘制为实数轴上方的半圆。在本文中，我们研究了最大线性排列问题（$\texttt{MaxLA}$），即$\texttt{minLA}$的最大化变体。我们提出了一种关于一般图最大排列的新刻画，并证明$\texttt{MaxLA}$可以在常数时间内解决圈图，且对于$k$线性树（$k\le2$）可在$O(n)$时间内解决。我们介绍了$\texttt{MaxLA}$的两个约束变体，分别称为$\texttt{二分图MaxLA}$和$\texttt{1-蓟MaxLA}$。我们证明，对于任何二分图，前者可在$O(n)$时间内解决；而后者通过一种通常在无标号树上运行时间为$O(n^3)$的算法解决。这两种变体的结合具有两个有前景的特点。首先，它对于几乎全部由数十个节点组成的树解决了$\texttt{MaxLA}$。其次，当算法未能求解$\texttt{MaxLA}$时，它对树产生了高质量近似解。此外，我们推测$\texttt{二分图MaxLA}$至少能解决$50\%$的自由树上的$\texttt{MaxLA}$问题。