On The Maximum Linear Arrangement Problem for Trees
Lluís Alemany-Puig, Juan Luis Esteban, Ramon Ferrer-i-Cancho
TL;DR
This work advances the study of Maximum Linear Arrangement (MaxLA) on trees by introducing a new characterization of maximum arrangements and dividing the problem into bipartite and non-bipartite components. It provides polynomial-time solutions for MaxLA on cycle graphs and, more importantly, on k-linear trees with k≤2, through a combination of maximal bipartite arrangements and carefully constructed non-bipartite arrangements (including 1-thistle MaxLA). The paper also presents two constrained variants—bipartite MaxLA and 1-thistle MaxLA—that together solve MaxLA for almost all trees and yield a 3/2-approximation, supported by empirical data. The results offer practical algorithms with strong performance on many trees and open directions for extending polynomial-time solvability and tightening approximation guarantees in broader graph classes.
Abstract
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.