Introduced by Papadimitriou and Yannakakis in 1989, layered graph traversal is an important problem in online algorithms and mobile computing that has been studied for several decades, and which now is essentially resolved in its original formulation. In this paper, we demonstrate that what appears to be an innocuous modification of the problem actually leads to a drastic (exponential) reduction of the competitive ratio. Specifically, we present an algorithm that is $O(\log^2 w)$-competitive for traversing unweighted layered graphs of width $w$. Our technique is based on a simple entropic regularizer, which evolves as the agent progresses in the layered graph. Our algorithm is randomized and simply maintains that at all layers, the probability distribution of the position of the mobile agent maximizes the entropic regularizer.

翻译：由Papadimitriou和Yannakakis于1989年提出的分层图遍历问题，是在线算法和移动计算领域的一个重要问题，经过数十年的研究，其原始表述已基本得到解决。本文证明，对该问题进行看似无伤大雅的修改，实际上会导致竞争比发生剧烈（指数级）下降。具体而言，我们提出了一种针对宽度为$w$的非加权分层图、具有$O(\log^2 w)$竞争比的算法。我们的技术基于一个简单的熵正则化器，该正则化器会随着智能体在分层图中前进而演化。该算法是随机化的，只需确保在所有分层上，移动智能体位置的概率分布能最大化该熵正则化器。