Maximal Linear Arrangement is NP complete on Bipartite Graphs

Abstract

The linear arrangement of a simple graph $G = (V,E)$ is a bijective labeling $π$ of the vertices of $G$ to the set $\{1, \ldots, n\}$. The maximum linear arrangement ($\texttt{maxLA}$) problem is to determine whether $G$ has a labeling such that the sum of labels across $G$'s edges, $\sum_{uv \in E} |π(u) - π(v)|$, is at least a given value $k$. Minimum linear arrangement ($\texttt{minLA}$) is defined analogously. Both $\texttt{maxLA}$ and $\texttt{minLA}$ are known to be NP complete for general graphs. $\texttt{minLA}$ has shown to be NP complete for bipartite graphs by Evans and Shiloah in 1975. It has been an open problem whether the same holds for $\texttt{maxLA}$. In this paper we prove that $\texttt{maxLA}$ is NP complete for bipartite graphs.