The Maximum Linear Arrangement Problem for trees under projectivity and planarity

TL;DR

This paper introduces linear-time algorithms for maximum linear arrangement problems on trees under planarity and projectivity constraints, providing new theoretical insights and extremal results for specific tree classes.

Contribution

It presents the first linear-time algorithms for MaxLA on trees with planarity and projectivity constraints, and characterizes extremal trees maximizing planar MaxLA.

Findings

Algorithms for planar and projective MaxLA are linear in time and space.

Caterpillar trees maximize planar MaxLA among all trees of fixed size.

Several properties of maximum arrangements are established.

Abstract

A linear arrangement is a mapping $\pi$ from the $n$ vertices of a graph $G$ to $n$ distinct consecutive integers. Linear arrangements can be represented by drawing the vertices along a horizontal line and drawing the edges as semicircles above said line. In this setting, the length of an edge is defined as the absolute value of the difference between the positions of its two vertices in the arrangement, and the cost of an arrangement as the sum of all edge lengths. Here we study two variants of the Maximum Linear Arrangement problem (MaxLA), which consists of finding an arrangement that maximizes the cost. In the planar variant for free trees, vertices have to be arranged in such a way that there are no edge crossings. In the projective variant for rooted trees, arrangements have to be planar and the root of the tree cannot be covered by any edge. In this paper we present algorithms that are linear in time and space to solve planar and projective MaxLA for trees. We also prove several properties of maximum projective and planar arrangements, and show that caterpillar trees maximize planar MaxLA over all trees of a fixed size thereby generalizing a previous extremal result on trees.