为图设计毛毛虫图：近似算法与难度
Designing Caterpillars for Graphs: Approximation and Hardness
AI总结：
该研究将最小线性排列（MLA）问题泛化为最大度不超过Δ的毛毛虫图设计问题，给出近似算法并证明其NP难性，还针对树的情况推导了4近似算法。
AI中文摘要：
经典的最小线性排列（Minimum Linear Arrangement, MLA）问题已被广泛研究，已知其为NP难问题，且存在O(√log n log log n)的近似算法[Feige和Lee，IPL，2007]。MLA可定义为如下设计问题：给定顶点集为V(G)的图G，设计一个顶点集相同的路径H，使线性排列代价∑_{uv∈E(G)} dist_H(u,v)最小，其中dist_H(u,v)表示H中u与v的距离。我们启动对该问题泛化形式的研究：允许H为最大度不超过Δ的毛毛虫图。毛毛虫图是路径的最简泛化，具有路径宽度1，且通过度参数Δ在路径与星型图之间插值。我们提出一种算法，可将MLA的任意α近似算法提升为我们问题的(α+3-2/(Δ-1))近似算法，从而为该更一般问题也得到O(√log n log log n)的近似算法。此外，当MLA可在多项式时间内求解时，我们推导得到一个4近似算法，尤其适用于树。作为上述结果的补充，我们证明对于每个常数Δ≥2，该问题均为NP难；且与MLA形成鲜明对比的是，当Δ为输入的一部分时，该问题在树中仍保持NP难。
英文摘要：
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 $Δ$. Caterpillars are the simplest generalization of paths, having pathwidth one and interpolating between paths and stars via the degree parameter $Δ$. We give an algorithm that lifts any $α$-approximation for MLA to an $(α+3-2/(Δ-1))$-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 $Δ\geq 2$, and, in stark contrast to MLA, show it remains NP-hard on trees when $Δ$ is part of the input.
