Maximum Linear Arrangement: exact algorithms for specific classes of graphs and approximation algorithms for wide classes of graphs
For researchers in graph algorithms and combinatorial optimization, this work offers new exact solutions and approximation guarantees for a previously less-studied problem, though results are limited to restricted graph classes.
This paper studies the Maximum Linear Arrangement problem (MaxLA), providing exact polynomial-time algorithms for specific graph classes (k-regular graphs with k≤2, k-linear trees with k≤2, and connected bipartite graphs) and a 3/2-approximation algorithm for trees via a constrained variant.
Linear arrangements of graphs are a well-known type of graph labeling and are found in many important computational problems. 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. A well-known computational problem is the Minimum Linear Arrangement Problem (${\tt minLA}$) where the goal is to find an arrangement that minimizes the sum of edge lengths. In this paper we study the Maximum Linear Arrangement problem (${\tt MaxLA}$), the counterpart of ${\tt minLA}$. We devise a new characterization of maximum arrangements of general graphs, and prove that ${\tt MaxLA}$ can be solved for $k$-regular graphs ($k\le2$) in time $O(n)$, and for $k$-linear trees ($k\le2$) in time $O(n)$. We present two constrained variants of ${\tt MaxLA}$ we call ${\tt bipartite MaxLA}$ and ${\tt 1-thistle MaxLA}$. We prove that the former can be solved in time $O(n)$ for any connected bipartite graph; the latter can be solved by an algorithm that typically runs in time $O(n^3\log n)$ on unlabeled trees. We show that ${\tt bipartite MaxLA}$ is a $3/2$-approximation algorithm for ${\tt MaxLA}$ for trees.