In the well-known Minimum Linear Arrangement problem (MinLA), the goal is to arrange the nodes of an undirected graph into a permutation so that the total stretch of the edges is minimized. This paper studies an online (learning) variant of MinLA where the graph is not given at the beginning, but rather revealed piece-by-piece. The algorithm starts in a fixed initial permutation, and after a piece of the graph is revealed, the algorithm must update its current permutation to be a MinLA of the subgraph revealed so far. The objective is to minimize the total number of swaps of adjacent nodes as the algorithm updates the permutation. The main result of this paper is an online randomized algorithm that solves this online variant for the restricted cases where the revealed graph is either a collection of cliques or a collection of lines. We show that the algorithm is $8 \ln n$-competitive, where $n$ is the number of nodes of the graph. We complement this result by constructing an asymptotically matching lower bound of $\Omega(\ln n)$.

翻译：在著名的最小线性排列问题（MinLA）中，目标是将一个无向图的节点排列成一个排列，使得所有边的总拉伸长度最小。本文研究MinLA的一种在线（学习）变体，其中图并非一开始就给定，而是逐步被揭示。算法从一个固定的初始排列开始，在图的一部分被揭示后，算法必须更新其当前排列，使其成为迄今为止揭示的子图的一个MinLA。目标是在算法更新排列的过程中，最小化相邻节点交换的总次数。本文的主要结果是针对揭示图是团集合或线集合的限制情况，提出了一种解决此在线变体的在线随机算法。我们证明了该算法是$8 \ln n$-竞争的，其中$n$是图的节点数。我们通过构造一个渐近匹配的$\Omega(\ln n)$下界来补充这一结果。