Shortest Paths without a Map, but with an Entropic Regularizer

TL;DR

This work tackles the online layered graph traversal problem, seeking short online paths in graphs revealed layer by layer. It reframes the problem through an evolving-tree game and solves it with online mirror descent using an entropic regularizer, achieving an $O(k^2)$-competitive randomized algorithm (up to a $O(\log d_{\max})$ factor in general). The analysis hinges on a potential combining a multiscale KL-divergence with a weighted depth metric, and it carefully handles topology changes via revised weights and damping in the control, leading to tight bounds that match the latest $\Omega(k^2)$ lower bound. By reducing layered graph traversal to evolving trees and then to small-set chasing, the paper links a broad class of online decision problems to a common, dynamically evolving geometric framework, with implications for metrical task systems and $k$-server-type problems.

Abstract

In a 1989 paper titled "shortest paths without a map", Papadimitriou and Yannakakis introduced an online model of searching in a weighted layered graph for a target node, while attempting to minimize the total length of the path traversed by the searcher. This problem, later called layered graph traversal, is parametrized by the maximum cardinality $k$ of a layer of the input graph. It is an online setting for dynamic programming, and it is known to be a rather general and fundamental model of online computing, which includes as special cases other acclaimed models. The deterministic competitive ratio for this problem was soon discovered to be exponential in $k$, and it is now nearly resolved: it lies between $Ω(2^k)$ and $O(k2^k)$. Regarding the randomized competitive ratio, in 1993 Ramesh proved, surprisingly, that this ratio has to be at least $Ω(k^2 / \log^{1+ε} k)$ (for any constant $ε> 0$). In the same paper, Ramesh also gave an $O(k^{13})$-competitive randomized online algorithm. Between 1993 and the results obtained in this paper, no progress has been reported on the randomized competitive ratio of layered graph traversal. In this work we show how to apply the mirror descent framework on a carefully selected evolving metric space, and obtain an $O(k^2)$-competitive randomized online algorithm. This matches asymptotically an improvement of the aforementioned lower bound (Bubeck, Coester, Rabani; STOC 2023), which we announced (among other results) after the initial publication of the results here.

Figures (1)

• Figure 1: Three types of metric and topology changes of the tree.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• Theorem 2
• Lemma 3
• Proof 1
• Lemma 4
• Proof 2
• Lemma 5