Designing Caterpillars for Graphs: Approximation and Hardness
The classical Minimum Linear Arrangement (MLA) problem has been studied extensively. It is known to be NP-hard and it admits an $O(\sqrt{\log n}\log\log n)$-approximation [Feige and Lee, IPL, 2007]. MLA can be defined as follows as design problem: Given a graph $G$ with vertex set $V(G)$, design a path $H$ on the same vertex set that minimizes the linear arrangement cost $\sum_{uv\in E(G)}\textrm{dist}_H(u,v)$, where $\textrm{dist}_H(u,v)$ indicates the distance of $u$ and $v$ in $H$. We initiate the study of the generalization in which $H$ is allowed to be a caterpillar graph of maximum degree at most $\Delta$. Caterpillars are the simplest generalization of paths, having pathwidth one and interpolating between paths and stars via the degree parameter $\Delta$. We give an algorithm that lifts any $\alpha$-approximation for MLA to an $(\alpha+3-2/(\Delta-1))$-approximation for our problem, thus obtaining an $O(\sqrt{\log n}\log\log n)$-approximation for our more general problem as well. Moreover, we derive a $4$-approximation whenever MLA is polynomial-time solvable, in particular, for trees. Complementing these results, we prove NP-hardness for every constant $\Delta\geq 2$, and, in stark contrast to MLA, show it remains NP-hard on trees when $\Delta$ is part of the input.