Unweighted Layered Graph Traversal: Passing a Crown via Entropy Maximization

TL;DR

This paper introduces a novel algorithm for layered graph traversal that significantly improves competitive ratios by leveraging entropy maximization, demonstrating exponential reductions in competitive bounds for unweighted layered graphs.

Contribution

It presents the first entropy-based approach to layered graph traversal, achieving an $O( ext{log}^2 w)$ competitive ratio for unweighted layered graphs.

Findings

Achieves exponential reduction in competitive ratio

Uses entropy maximization for decision-making

Provides theoretical bounds for unweighted layered graphs

Abstract

Introduced by Papadimitriou and Yannakakis in 1989, layered graph traversal is a central 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 algorithm chooses the agent's position simply according to the probability distribution over the current layer that maximizes the sum of entropies of the induced distributions in the preceding layers.